Chapter 3 Differentiation
微积分常用英文词汇(分章)
英汉微积分词汇English-Chinese Calculus Vocabulary第一章函数与极限Chapter 1 Function and Limit高等数学higher mathematics集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product象限quadrant原点origin坐标coordinate轴axisx 轴x-axis整数integer有理数rational number实数real number开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心center of a neighborhood领域的半径radius of a neighborhood左领域left neighborhood右领域right neighborhood映射mappingX到Y的映射mapping of X onto Y满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function单值分支one-valued branch函数图形graph of a function绝对值absolute value绝对值函数absolute value function符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if (iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness最小上界least upper bound无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing严格递减strictly decreasing严格递增strictly increasing单调函数monotone function函数的奇偶性parity (odevity) of a function 对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period周期函数periodic function反函数inverse function直接函数direct function函数的复合composition of function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function线性函数linear function常数函数constant function多项式polynomial分段定义函数piecewise defined function阶梯函数step function幂函数power function指数函数exponential function指数exponent自然指数函数natural exponential function对数logarithm对数函数logarithmic function自然对数函数natural logarithm function三角函数trigonometric function正弦函数sine function余弦函数cosine function正切函数tangent function半角公式half-angle formulas反三角函数inverse trigonometric function常数函数constant function双曲线hyperbola双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent最优化问题optimization problems不等式inequality极限limit数列sequence of number复利compound interest收敛convergence收敛的convergent收敛于a converge to a发散divergence发散的divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列 subsequence函数的极限 limits of functions函数当x 趋于0x 时的极限 limit of functions as x approaches 0x单侧极限 one-sided limit左极限 left limit右极限 right limit单侧极限 one-sided limits渐近线 asymptote水平渐近线 horizontal asymptote分式 fractions商定律 quotient rule无穷小 infinitesimal无穷大 infinity铅直渐近线 vertical asymptote夹逼准则 squeeze rule (Sandwich theorem)单调数列 monotonic sequence高阶无穷小 infinitesimal of higher order低阶无穷小 infinitesimal of lower order同阶无穷小 infinitesimal of the same order等阶无穷小 equivalent infinitesimal多项式的次数 degree of a polynomial三次函数 cubic function函数的连续性 continuity of a function增量 increment函数在0x 连续 the function is continuous at 0x左连续 left continuous / continuous from the left右连续 right continuous / continuous from the right连续性 continuity不连续性 discontinuity连续函数 continuous function函数在区间上连续 function is continuous on an interval不连续点 discontinuity point第一类间断点 discontinuity point of the first kind第二类间断点 discontinuity point of the second kind初等函数的连续性 continuity of the elementary functions定义区间 defined interval最大值 global maximum value (absolute maximum)最小值 global minimum value (absolute minimum)零点定理 the zero point theorem介值定理 intermediate value theorem第二章 导数与微分Chapter 2 Derivative and Differential 速度velocity速率speed平均速度average velocity瞬时速度instantaneous velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle割线secant line切线tangent line位置函数position function导数derivative求导法differentiation可导的derivable可导函数differentiable function光滑曲线smooth curve变化率rate of change函数的变化率问题problem of the change rate of a function导函数derived function导数定义域domain of derivative左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives在闭区间[a, b]上可导is derivable on the closed interval [a, b] 指数函数的导数derivative of exponential function幂函数的导数derivative of power function切线的斜率slope of the tangent line截距interceptsx 截距x-intercept直线的斜截式slope-intercept equation of a line点斜式point-slope form切线方程tangent equation焦点focus角速度angular velocity成本函数cost function边际成本marginal cost逐项求导法differentiation term by term积之导数derivative of a product商之导数derivative of a quotient链式法则chain rule隐函数implicit function显函数explicit function隐函数求导法implicit differentiation加速度acceleration二阶导数second derivative三阶导数third derivative高阶导数nth derivative / higher derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数parameter参数方程parametric equation相关变化率correlative change rata微分differential微分学differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient逼近法approximation用微分逼近approximation by differentials间接测量误差indirect measurement error绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter 3 MeanValue Theorems of Differentials and the Application ofDerivatives均值定理mean value theorem罗尔定理Roll’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule不定式indeterminate form“0”型不定式indeterminate form of type “”泰勒中值定理Taylor’s mean value theorem 泰勒公式Taylor formula系数coefficient余项remainder term线性近似linear approximation拉格朗日余项Lagrange remainder term麦克劳林公式Maclaurin’s formula佩亚诺余项Peano remainder term阶乘factorial凹凸性concavity上凹（或下凸）concave upward (concave up)下凹（或向上凸的）concave downward (concave down) 拐点inflection point极值extreme value函数的极值extremum of function极大与极小值maximum and minimum values极大值local (relative) maximum最大值global (absolute) maximum极小值local (relative) minimum最小值global (absolute) minimum目标函数objective function收入函数revenue function斜渐进线slant asymptote曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter 4 Indefinite Integrals 原函数primitive function / antiderivative积分integration积分学integral积分号sign of integration被积函数integrand积分变量integral variable积分常数constant of integration积分曲线integral curve积分表table of integrals换元积分法integration by substitution有理代换法rationalizing substitution三角代换法trigonometric substitutions分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function真分式proper fraction假分式improper fraction部分分式partial fractions三角积分trigonometric integrals第五章定积分Chapter 5 Definite Integrals曲线下方之面积area under a curve曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition黎曼和Riemannian sum积分和integral sum可积的integrableSimpson 法则逼近法approximation by Simpson’s rule梯形法则逼近法approximation by trapezoidal rule矩形法rectangle method曲线之间的面积area between curves积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an interval 牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus微积分基本定理fundamental theorem of calculus变量代换change of variable换元公式formula for integration by substitution递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions瑕点flaw绝对收敛absolutely convergent第六章定积分的应用Chapter 6 Applications of the Definite Integrals 元素法the element method面积元素element of area平面plane平面图形的面积area of a plane figure直角坐标（又称“笛卡儿坐标”）Cartesian coordinates / rectangular coordinates x 坐标x-coordinate坐标轴coordinate axes极坐标polar coordinates极轴polar axis极点pole圆circle扇形sector抛物线parabola椭圆ellipse椭圆的轴axes of ellipse蚌线conchoid外摆线epicycloid双纽线lemniscate蚶线limacon旋转体solid of revolution, solid of rotation旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of a curve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter 7 Space Analytic Geometry and Vector Algebra纯量（标量）scalar向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel平行线parallel lines向量的线性运算linear operation of vector加法addition减法subtraction数乘运算scalar multiplication三角形法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law分配律distributive law负向量negative vector三角不等式triangle inequality对角线diagonal差difference余弦定理（定律）law of cosines空间space空间直角坐标系space rectangular coordinates坐标平面coordinate plane卦限octant向量的模modulus of vector定比分点definite proportion and separated point中点公式midpoint formula等腰三角形isosceles triangle向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle投影projection向量在轴上的投影projection of a vector onto an axis向量的分量components of a vector对称点symmetric point数量积（点积，内积）scalar product (dot product, inner product)叉积（向量积，外积）cross product (vector product, exterior product) 混合积mixed product锐角acute angle流体fluid刚体rigid body角速度angular velocity平行六面体parallelepiped平面的点法式方程point-norm form equation of a plane法向量normal vector平面的一般方程general form equation of a plane三元一次方程three-variable linear equation平面的截距式方程intercept form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程point-direction form equations of a line直线的对称式方程symmetric form equation of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular垂直线perpendicular lines直线与平面的夹角angle between a line and a planes 平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant曲面方程equation for a surface球面sphere球体spheroid球心ball center轨迹方程locus equation旋转轴rotation axis旋转曲面surface of revolution母线generating line圆锥面cone顶点vertex半顶角semi-vertical angle旋转双曲面revolution hyperboloids旋转单叶双曲面revolution hyperboloids of one sheet 旋转双叶双曲面revolution hyperboloids of two sheets 柱面cylindrical surface, cylinder圆柱circular cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface截痕法method of cut-off mark椭圆锥面elliptic cone椭球面ellipsoid双曲面hyperboloid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution抛物面paraboloid椭圆体ellipsoid椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve交线intersection curve空间曲线的一般方程general form equations of a space curve空间曲线的参数方程parametric equations of a space curve螺线spiral / helix螺矩pitch投影柱面projecting cylinder第八章多元函数微分法及其应用Chapter 8 Differentiation of Functions of Several Variables and Its Application 一元函数function of one variable二元函数binary function邻域neighborhood去心邻域noncentral neighborhood方邻域square neighborhood圆邻域circular neighborhood内点interior point外点exterior point边界点frontier point, boundary point聚点point of accumulation导集derived set开集open set闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimensional space多元函数function of several variables二重极限double limit多元函数的连续性continuity of function of several variables连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative混合偏导数hybrid partial derivative全微分total differential偏增量partial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superposition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function 光滑曲面smooth surface曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative等高线level curve梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values鞍点saddle point条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter 9 Multiple Integrals重积分multiple integrals二重积分double integral可加性additivity累次（逐次）积分iterated integral体积元素volume element二重积分变量代换法change of variable in double integral极坐标表示的面积area in polar coordinates扇形的面积area of a sector of a circle极坐标二重积分double integral in polar coordinates三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system 柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system 球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system 剥壳法shell method圆盘法disk method反常二重积分improper double integral曲面的面积area of a surface质心center of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter 10 Line (Curve) Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc length第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x, y, and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region路径无关path independence格林公式Green formula顺时针方向clockwise逆时针方向counterclockwise区域边界的正向positive direction of region boundary第一类曲面积分surface integrals of the first type旋转曲面的面积area of a surface of a revolution对面积的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面之元素element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator拉普拉斯变换Laplace transform格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation (curl)第十一章无穷级数Chapter 11 Infinite Series一般项general term部分和partial sum收敛级数convergent series余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence 正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coefficients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence幂级数的导数derivative of power series泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error (rounding error)欧拉公式Euler’s formula魏尔斯特拉斯判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立叶级数Fourier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter 12 Differential Equation解微分方程solve a differential equation常微分方程ordinary differential equation (ODE)偏微分方程partial differential equation (PDE)微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解implicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient state current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linear differential equation of higher order自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linear differential equation线性相关linearly dependence线性无关linearly independence二阶常系数齐次线性微分方程second order homogeneous linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping固有频率natural frequency简谐振动simple harmonic oscillation, simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerical solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linear differential equations with constant coefficient。
AP微积分课程设计
北京中加学校AP微积分课程的实行方案正当北京中加学校建校十五周年之际，为了实现百年名校的伟大目标，不论是在管理，仍是在教育教课上都需要不停健全和完美体系体制。
但是，学科课程的建设和更新必然首当其冲，火烧眉毛。
暑期在大连举行的《北京中加学校数学课程和教课多元化的研究》教学商讨会为北京中加学校的学科建设开了先河，也确立了思想基础。
借此良机，对于北京中加学校的特点学科之一，AP 微积分，我们借鉴过去的教课经验，整合国内外教课资源，依照美国大学理事会AP微积分的课程标准，制定了对于AP微积分课程的教课假想。
一、指导思想本课程是北京中加学校为学生开设的一门国际数学专业基础课。
开设本课程的目的，在于以美国大学理事会规定的 AP?微积分课程标准为指导，依照理论与实践相联合的原则，经过对微积分基来源理及规律的讲解，使学生系统掌握极限、连续、导数和积分等知识的基来源理、基本内容和基本方法，对微积分在经济活动中的应用有比较清楚的认识，提升学生专业词汇量和阅读英语原版书本的能力，拓宽学生国际数学视线，使学生体验到数学的价值和美学认知。
课内学时 144，4 学分，从高一第一学期开始开设，高二第二学期结束，快要两个学年授完。
二、课程目标AP 微积分是在高中学习阶段有余力、有能力、成绩优异的学生有时机先修的美国大学基础课程以获取美国大学学分专业的必修课。
要修业生在学完本课程后，掌握本课程的基来源理、基本内容、基本方法及基本知识，并拥有对所学的微积分知识进行现实理解和实质应用的能力，进而顺利经过AP考试。
据此，本课程查核侧重于基本知识的掌握、理解和应用剖析能力两个方面。
在各章的查核要求中，相关基本观点、基本理论、基本公式、应用剖析能力的内容按“识记、理解、简单应用和综合应用”四个层次要求。
三、教课进度北京中加学校 AP微积分教课内容及其进度计划普学期国际班AP微积分课时分派通班第第会合（ 4 课时）复合函数、反函数以高中课程： 6课高一一及作图计算器的使用时/ 周，共 54课函数与基本初等函数时；一学模（32 课时）期块AP 微积分： 2课分析几何（ 9 课时）时/ 周，共 18课时；第直线与圆的方程（9极限高中课程： 6课时/ 周，共 54课二课时）极限的运算法例时；模圆锥曲线（ 8 课时）块AP 微积分： 2课三角函数（ 16 课时）时/ 周，共 18课时；第第三角恒等变换导数高中课程： 2课时/ 周，共 18课二三反三角函数导数的基本公式学模时；期块AP 微积分： 6课时/ 周，共 54课时；第立体几何导数的运算法例高中课程： 2课四时/ 周，共 18课模时；块AP 微积分： 6课时/ 周，共 54课时；第第常用逻辑用语导数的应用高中课程： 2 课高五时/ 周，共18 课一平面向量模时；二学块解三角形AP 微积分： 6 课期时/ 周，共54 课时；第数列积分方程高中课程： 2 课六不等式微分方程时/ 周，共 18课模时；块AP 微积分： 6课时/ 周，共 54课时；第第复数AP微积分总复习高中课程： 2课二七统计AP微积分 AB考试时/ 周，共 18课模时；学块计数原理AP 微积分： 6课期时/ 周，共 54课时；第参数方程AP微积分 BC高中课程： 4课八时/ 周，共 36课极坐标模时；块AP 微积分 BC：4课时 / 周，共36课时；第总总复习AP微积分 BC高中课程： 4 课高时/ 周，共36 课一复毕业会考三时；习学AP 微积分 BC：4期课时 / 周，共课时；36第微其余大学预修课程AP微积分 BC高中课程： 4课二积AP微积分 BC考试时/ 周，共 36课分时；学AP 微积分 BC：4期课时 / 周，共36课时；四、课程内容Chapter 2 Limits and Derivatives第二章极限和导数Teaching Teaching Requirements and Objectives TimeContent教课要乞降目标学时教课内容The Tangent The student will apply the derivativeand Velocity to solve problems,including tangentProblems and normal lines to a curve,curve切线和速率问题sketching, velocity, acceleration.The Limit of a Function The student will define and applyproperties of limits of functions.the函数的极限This will include limits of a constant, Calculatingsum,product,quotient,one-sidedlimits,limits at infinity,infinite Limits Usinglimits, and nonexistent limits.the Limit Laws利用极限法例计算极限Continuity The student will state the definitionof continuity and determine where a 连续性function is continuous ordiscontinuous.This will include continuity at apoint;continuity over a closedinterval;and graphical interpretationof continuity and discontinuity.Limits at Infinity; Horizontal Asymptotes The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these无量远处极限和水平渐近线Tangents, Velocities,and Other Rates of Change切线、速度和其它的变化率functions using a graphing calculator. Properties of functions will includedomains, ranges, combinations, odd,even, periodicity,symmetry, asymptotes, zeros, upper andlower bounds, and intervals where thefunction isincreasing or decreasing.The student will also define and applytheproperties of limits of functions.This will include limits of a constant,sum,product,quotient,one-sided limits,limits atinfinity,infinite limits,and nonexistent limits.Derivatives The student will find the derivative of导数an algebraic function by using thedefinition of a derivative.The Derivativeas a Function This will include investigating anddescribing the relationship between导函数differentiability and continuity.Review复习Chapter 3 Differentiation Rules第三章导数法例Teaching Teaching Requirements and Objectives TimeContent教课要乞降目标学时教课内容Derivatives of The student will apply formulas to findPolynomials and the derivative of algebraic, Exponential trigonometric,exponential,and Functions logarithmic functions and theirinverses.多项式函数和指数函数的导数The Product The student will apply formulas to findand Rules Quotient the derivative of the sum of elementary functions.导数的乘法和除法运算法例Rates of Students will be able to understand the Change in the mathematical modeling process of Natural and derivatives(rates of changes) in the Social Sciences real world自然科学和社会科学中的变化率of Trigonometric Students willdifferentiationberulesableofto use thetrigonometricFunctions functions三角函数的导数The Chain Rule The student will apply formulas to find链式法例the derivative of the sum, product,quotient, inverse and composite (chainrule) of elementary functions.Implicit The student will find the derivative ofan implicitly defined function.Differentiation隐函数求导Higher The student will find the higher orderderivatives of algebraic, Derivativestrigonometric,exponential,and 高阶导数logarithmic functions.of Logarithmic The student will use logarithmic Functions differentiation as a technique todifferentiate non-logarithmic对数函数的导数functions.Hyperbolic The student will be able to understand Functions the definition of hyperbolic functions,and solve for its derivatives.双曲函数Linear The student will apply the derivative Approximations to solve problems,including tangent and and normal lines to a curve, curve Differentials sketching,velocity,acceleration,related rates of change,Newton's 线性迫近和微分method,differentials and linearapproximations,and optimizationproblems.Review复习Chapter 4 Applications of Differentiation第四章导数的应用Teaching Teaching Requirements and Objectives TimeContent教课要乞降目标学时教课内容Maximum and The student will be able to understandMinimum Values extreme values of a function, find极大值和极小值critical values of a function and findextreme values of a function.The Mean Value The student will state (without proof) Theorem the Mean Value Theorem for derivatives中值定理and apply it both algebraically andgraphically.How Derivative Affect the Shape of a The student will graph these functions using a graphing calculator, including understanding and using the FirstGraph Derivative Test and the Second导数是怎样改变Derivative Test to determine min’s andmax’s.图像的形状Indeterminate The student will use l'Hopital's rule Forms and L ’to find the limit of functions whose Hospital ’s limits yield the indeterminate forms: Rules0/0 and infinity/infinity不定式和洛必达法例Optimization Problems The student will be able to use derivatives to solve optimization problems最优化问题Newton ’s The student will be able to use Method Newton’s method to approximate roots 牛顿法例of an equation.The student will be able to understandthe concept of an antiderivative, the原函数（反导geometry of the antiderivative and that 数）of slope fields and also workrectilinear motion problems withantiderivativesReview复习Chapter 5 Integrals第五章积分Teaching Teaching Requirements and Objectives TimeContent教课要乞降目标学时教课内容Areas and Distances The student will identify properties ofthe definite integral.the面积和距离This will include the FundamentalTheorem of Calculus and the definiteintegral as an area and as a limit of asum as well as the fundamental theorem.The Definite Integral The student will compute an approximate value for a definite integral.不定积分This will include numericalcalculations using Riemann Sums and theTrapezoidal Rule.The The student will identify the Fundamental properties of the definite integral. Theorem ofThis will include the Fundamental CalculusTheorem of Calculus and the definite微积分基本定理integral as an area and as a limit of asum as well as the fundamental theorem.The integral from a to x of f(t)d(t)dt/dx = f(x)Indefinite Integrals and the Net Change Theorem The student will find the indefinite integral of algebraic, exponential, logarithmic, and trigonometric functions.不定积分和原函数定理The Substitution Rule The student will find the indefinite integral of algebraic, exponential, logarithmic, and trigonometric functions.换元积分法The special integration techniques ofsubstitution (change of variables) andintegration by parts will be included.Review复习Chapter 6 Applications of Integration第六章积分的应用Teaching Teaching Requirements and Objectives Time Content教课要乞降目标学时教课内容Areas Between Curves The student will applyintegral to solve problems.the definite曲边面积These problems will include findingdistance traveled on a line andvelocity from acceleration with initialconditions,growth and decay problems,solutions of separable differentialequations,the average value of afunction,area between curves,volumesof solids of revolution about the axesor lines parallel to the axes usingdisc/washer and shell methods,andvolumes of solids with known cross-sectional areas.Volumes The student will apply the definiteintegral to solve problems.体These problems will include areabetween curves, volumes of solids ofrevolution about the axes or linesparallel to the axes using disc/washerand shell methods, and volumes ofsolids with known cross-sectionalareas.Volumes by The student will apply the definite Cylindrical integral to solve problems.ShellsThese problems will include⋯ area 柱体体between curves, volumes of solids ofrevolution about the axes or linesparallel to the axes using disc/washerand shell methods, and volumes ofsolids with known cross-sectionalareas.Work The student will apply the definiteintegral to solve problems.物体功These problems will include findingdistance traveled on a line andvelocity from acceleration with initialconditions, growth and decay problems,work done.Average Value The student will apply the definiteof a Function integral to solve problems.实函数均值These problems will include the averagevalue of a function.Density The student will apply the definite Function integral to solve problems.密度函数These problems will include findingdistance traveled on a line andvelocity from acceleration with initialconditions, growth and decay problems. Review复习Chapter 7 Techniques of Integration第七章积分技巧Teaching Teaching Requirements and Objectives Time Content教课要乞降目标学时教课内容Integration by The student will find the definite andParts indefinite integral of algebraic,exponential, logarithmic, and分部积分法trigonometric functions.TrigonometricThe special integration techniques ofIntegralssubstitution (change of variables) and三角函数积分integration by parts will be included.TrigonometricSubstitution三角函数替代Integration ofRationalFunctions byPartialFractions有理函数的分部积分Review复习五、查核方式为了彰显我校“多一把尺子，多一位人材”的教育教课理念，本课程采纳形成性查核与终结性查核相联合的方式。
台湾国立交通大学
台湾国⽴交通⼤学数学视频数学视频Calculus I 台湾国⽴交通⼤学 Michael Fuchs⽼師 36集（点击进⼊我的淘宝店）Calculus II 台湾国⽴交通⼤学 Michael Fuchs⽼師 29集（点击进⼊我的淘宝店）Chapter1 Functions and Model1-5 Exponential Functions1-6 Inverse Functions and LogarithmsChapter2 Limits and Derivatives2-2 The Limit of a Function2-4 The Precise Definition of a Limit2-3 Calculating Limits Using the Limit Laws2-6 Limits at Infinity; Horizontal Asymptotes2-5 Continuity2-8 Derivatives2-9 The Derivative as a FunctionChapter3 Differentiation Rules3-1 Derivatives of Polynomials and Exponential Functions3-2 The Product and Quotient Rules3-4 Derivatives of Trigonometric Functions3-5 The Chain Rule3-6 Implicit Differentiation3-8 Derivatives of Logarithmic Functions3-10 Related Rates3-7 Higher Derivatives3-11 Linear Approximations and DifferentialsChapter4 Applications of Differation4-1 Maximum and Minimum Values4-2 The Mean Value Theorem4-3 How Derivatives Affect the Shape of a Graph4-4 Indeterminate Forms a nd L’Hospital’s Rule4-7 Optimization Problems4-5 Summary of Curve Sketching4-10 AntiderivativesChapter5 Integrals5-1 Areas and Distances5-2 The Definite Integral5-3 The Fundamental Theorem of Calculus5-4 Indefinite Integrals and the Total Change Theorem5-5 The Substitution Rule5-6 The Logarithm Defined as an IntegralChapter6 Applications of Integration6-1 Areas between Curves6-2 Volumes6-3 Volumes be Cylindrical ShellsChapter7 Techniques of Integration7-1 Integration by Parts7-2 Trigonometric Integrals7-3 Trigonometric Substitution7-4 Integration of Rational Functions by Partial Fractions7-8 Improper Integrals7-7 Approximate IntegrationChapter8 Further Applications of Integration8-1 Arc Length8-2 Area of a Surface of RevolutionChapter10 Parametric Equations and Polar Coordinates10-1 Curves Defined by Parametric Equations10-2 Calculus with Parametric Curves10-3 Polar Coordinates10-4 Areas and Lengths in Polar Coordinates微积分(⼀) 台湾国⽴交通⼤学莊重⽼師 24集（点击进⼊我的淘宝店）微积分(⼆)台湾国⽴交通⼤学莊重⽼師 24集（点击进⼊我的淘宝店）課程章節第⼀章Functions and Model第⼆章Limits and derivatives第三章Differentiation Rules第四章The Properties of Gases第五章Integrals第六章Applications of Integration第七章Techniques of Integration第⼋章Further Applications of Integration第⼗章Parametric Equations and Polar Coordinates第⼗⼀章Infinite Sequences and Series第⼗⼆章Vectors and the Geometry of Space第⼗三章Vector Functions第⼗四章Partial Derivatives第⼗五章Multiple Integrals⾼等微积分(⼀)台湾国⽴交通⼤学⽩啟光⽼師 29集（点击进⼊我的淘宝店）⾼等微积分(⼆) 台湾国⽴交通⼤学 ⽩啟光⽼師 27集（点击进⼊我的淘宝店）第⼀章The Real and Complex Number SystemsFields Axioms, Order Axioms Completeness Axioms第⼆章Basic TopologyCardinality of SetsMetric SpacesCompact SetsConnected Sets第三章Numerical Sequences and SeriesConvergent SequencesCauchy SequencesUpper and Lower LimitsSeries of Nonnegative TermsThe Root and Ratio TestAbsolute Convergence, Rearrangements第四章ContinuityLimits of Functions and Continuous FunctionsContinuity and CompactnessContinuity and Connectednessdiscontinuities, Infinite Limits and Limits at Infinity第五章Differentiation The Derivative of a Real Function, Mean Value TheoremL’Hopital’s RuleTaylor’s TheoremDifferentiation of Vector-valued Functions第六章The Riemann-Stieltjes Integral Definition and Existence of the IntegralProperties of the IntegralIntegration and DifferentiationIntegration and Differentiation第六章The Riemann-Stieltjes IntegralIntegration and Differentiation第七章Sequence and Series of FunctionsSequence and Series of Functions --- the Main ProblemUniform Convergence and ContinuityUniform Convergence and IntegrationUniform Convergence and DifferentiationEquicontinuous Family of FunctionsThe Stone-Weierstrass Theorem第⼋章Some Special FunctionsPower seriesSome Special FunctionsFourier SeriesThe Gamma Function第九章Functions of several variablesFunction of Several VariablesFunction of Several Variables：DifferentiationFunction of Several Variables：DifferentiationThe Inverse Function TheoremThe Implicit Function TheoremThe Rank TheoremDeterminantsDifferentiation of Integrals偏微分⽅程(⼀) 台湾国⽴交通⼤学林琦焜⽼师 3.8GB （点击进⼊我的淘宝店）偏微分⽅程(⼆) 台湾国⽴交通⼤学林琦焜⽼师 3.4GB （点击进⼊我的淘宝店）内容纲要第⼀章 The Single First-Order Equation1-1 Introduction Partial differential equations occur throughout mathematics. In this part we will give some examples1-2 Examples1-3 Analytic Solution and Approximation methods in a simple example 1-st order linear example1-4 Quasilinear Equation The concept of characteristic1-5 The Cauchy Problem for the Quasilinear-linear Equations1-6 Examples Solved problems1-7 The general first-order equation for a function of two variables characteristic curves, envelope1-8 The Cauchy Problem characteristic curves, envelope1-9 Solutions generated as envelopes第⼆章Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables2-1 Characteristics for Linear and Quasilinear Second-Order Equations Characteristic2-2 Propagation of Singularity Characteristic curve and singularity2-3 The Linear Second-Order Equation classification of 2nd order equation2-4 The One-Dimensional Wave Equation dAlembert formula, dimond law, Fourier series2-5 System of First-Order Equations Canonical form, Characteristic polynominal2-6 A Quasi-linear System and Simple Waves Concept of simple wave第三章 Characteristic Manifolds and Cauchy Problem3-1 Natation of Laurent Schwartz Multi-index notation3-2 The Cauchy Problem Characteristic matrix, characteristic form3-3 Real Analytic Functions and the Cauchy-Kowalevski Theorem Local existence of solutions of the non-characteristic 3-4 The Lagrange-Green Identity Gauss divergence theorem3-5 The Uniqueness Theorem of Ho ren Uniqueness of analytic partial differential equations3-6 Distribution Solutions Introdution of Laurent Schwartzs theory of distribution (generalized function)第四章 The Laplace Equation4-1 Greens Identity, Fundamental Solutions, and Poissons Equation Dirichlet problem, Neumann problem, spherical symmetry, mean value theorem, Poisson formula4-2 The Maximal Principle harmonic and subharmonic functions4-3 The Dirichlet Problem, Greens Function, and Poisson Formula Symmetric point, Poisson kernel4-4 Perrons method Existence proof of the Dirichlet problem4-5 Solution of the Dirichlet Problem by Hilbert-Space Methods Functional analysis, Riesz representation theorem, Dirichlet integra第五章 Hyperbolic Equations in Higher Dimensions5-1 The Wave Equation in n-Dimensional Space(1) The method of sphereical means(2) Hadmards method of descent(3) Duhamels principle and the general Cauchy problem(4) mixed problem5-2 Higher-Order Hyperbolic Equations with Constant Coefficients(1) Standard form of the initial-value problem(2) solution by Fourier transform,(3) solution of a mixed problem by Fourier transform5-3 Symmetric Hyperbolic System(1) The basic energy inequality(2)Finite difference method(3) Schauder method第六章 Higher-Order Elliptic Equations with Constant Coefficients6-1 The Fundamental Solution for Odd n Travelling wave6-2 The Dirichlet Problem Lax-Milgram theorem, Garding inequality6-3 Sobolev Space Weak solution and Hibert space第七章 Parabolic Equations7-1 The Heat Equation Self-Similarity, Heat kernel, maximum principle7-2 The Initial-Value Problem for General Second-Order Parabolic Equations(1) Finite difference and maximum principle(2) Existence of Initial Value Problem第⼋章 H. Lewys Example of a Linear Equation without Solutions8-1 Brief introduction of Functional Analysis Hilbert and Banach spaces, projection theorem, Leray-Schauder theorem8-2 Semigroups of linear operator Generation, representation and spectral properties8-3 Perturbations and Approximations The Trotter theorem8-4 The abstract Cauchy Problem Basic theory8-5 Application to linear partial differential equations Parabolic equation, Wave equation and Schrodinger equation8-6 Applications to nonlinear partial differential equations KdV equation, nonlinear heat equation, nonmlinear Schrodinger equation变分学导论应⽤数学系林琦焜⽼师台湾国⽴交通⼤学 2GB （点击进⼊我的淘宝店）内容纲要第⼀章变分学之历史名题1.1 Bernoulli 最速下降曲线1.2 最⼩表⾯积的迴转体1.3 Plateau问题(最⼩曲⾯)1.4 等周长问题1.5 古典⼒学之问题第⼆章 Euler- Lagrange⽅程2.1 变分之原理2.2 折射定律与最速下降曲线2.3 ⼴义座标2.4 Dirichlet 原理与最⼩曲⾯2.5 Lagrange乘⼦与等周问题2.6 Euler-Lagrage ⽅程之不变量2.7 Sturm-Liouville问题2.8 极值(积分)问题第三章 Hamilton系统3.1 Legendre变换3.2 Hamilton⽅程3.3 座标变换与守恒律3.4 Noether定理3.5 Poisson括号第四章数学物理⽅程4.1 波动⽅程4.2 Laplace与Poisson⽅程4.3 Schrodinger ⽅程4.4 Klein-Gordon ⽅程4.5 KdV ⽅程4.6 流体⼒学⽅程 课程书⽬变分学导论 (Lecture note by Chi-Kun Lin).向量分析台湾国⽴交通⼤学林琦焜 3.3GB （点击进⼊我的淘宝店）向量分析主要是要谈”梯度、散度与旋度”这三个重要观念，⽽对应的则是⽅向导数、散度定理、与Stokes定理因此重⼼就在於如何釐清线积分、曲⾯积分以及他们所代表的物理意义。
高等数学 英文教材
高等数学英文教材Higher Mathematics: An English TextbookIntroduction:Higher Mathematics is a crucial subject for students pursuing degrees in STEM (Science, Technology, Engineering, and Mathematics) fields. This English textbook aims to provide a comprehensive and accessible resource for students studying higher mathematics in an international academic context. With a focus on clarity, logical presentation, and English language proficiency, this textbook will equip students with the necessary mathematical skills and knowledge to succeed in their academic journey.Chapter 1: Differentiation1.1 Fundamental ConceptsDifferentiation is a fundamental topic in calculus, enabling students to analyze the behavior of functions. This chapter will cover the basic rules of differentiation, including the power rule, product rule, quotient rule, and chain rule. Various examples and exercises will be provided to ensure students grasp the concepts effectively.1.2 Applications of DifferentiationBuilding upon the foundational concepts, this section explores the applications of differentiation. Students will learn how to find critical points, determine concavity and inflection points, optimize functions, and solve real-world problems using differentiation techniques. The significance ofdifferentiation in solving practical problems in disciplines such as physics, economics, and engineering will be highlighted.Chapter 2: Integration2.1 Definite and Indefinite IntegralsThis chapter focuses on the concept of integration. Students will explore indefinite integrals, fundamental theorem of calculus, and techniques such as substitution and integration by parts. The significance of integration in finding areas, volumes, and computing sums will be emphasized.2.2 Applications of IntegrationIn this section, students will delve into the various applications of integration. They will learn how to find the area between curves, calculate volumes of solids of revolution, and solve real-world problems using integration techniques. The importance of integration in physics, economics, and statistics will be demonstrated through examples and exercises.Chapter 3: Differential Equations3.1 First-order Differential EquationsThis chapter introduces students to first-order differential equations and their applications. The concepts of separable equations, linear equations, and Bernoulli equations will be covered. Students will gain an understanding of the fundamental techniques for solving differential equations.3.2 Second-order Differential EquationsExpanding on the previous section, this part focuses on second-order differential equations. Students will explore homogeneous and non-homogeneous equations, as well as various methods for solving them, including the method of undetermined coefficients and variation of parameters. Applications of second-order differential equations in physics and engineering will be discussed.Chapter 4: Sequences and Series4.1 Sequences and ConvergenceThis chapter introduces students to sequences and their convergence properties. The concepts of limits, convergence, and divergence will be explored. Students will learn how to analyze the behavior of sequences and determine their convergence using various tests.4.2 Series and ConvergenceBuilding upon the previous section, this part delves into series and their convergence properties. Students will study different types of series, including geometric, p-series, and alternating series. Convergence tests such as the comparison test, ratio test, and integral test will be covered. Practical applications of series in calculus and numerical methods will be discussed.Conclusion:This English textbook on Higher Mathematics provides a comprehensive and well-structured resource for students. Through its clear explanations, numerous examples, and practice exercises, students will develop a strong foundation in calculus, enabling them to tackle advanced mathematical problems confidently. With a focus on English language proficiency, this textbook caters to the needs of international students pursuing higher education in mathematics-related disciplines.。
(2020年整理)AP Calculus AB review AP微积分复习提纲.pptx
*The Derivative of the Natural Exponential Function Let u be a differentiable fu.
*Integration Rules for Exponential Functions Let u be a differentiable function of x.
*The Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f ’(c) =
area for each sub-interval and summation is the integral). *Definite integral *The Fundamental Theorem of Calculus
If a function f is continuous on the closed interval [a, b] and F is an anti-derivative of f on the interval [a, b], then
*Definition of the Hyperbolic Functions
学海无 涯
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). Definition of the Derivative of a Function The Derivative of f at x is given by
C1词汇表
Vocabulary for Core Mathematics IChapter 1 Algebra and functions1.algebra ['ældʒɪbrə] n. 代数学2.multiply ['mʌtɪplaɪ] v. 乘，使相乘，做乘法，使增加，增加，繁殖3.simplify ['sɪmplɪfaɪ] v. 简化，精简，使平易，单纯4.index ['ɪndeks] n. 索引，指数，指针v. 编入索引中，指出mon ['kɒmən] adj. 通常的，公共的，通俗的n. 公共用地，公有地6.quadratic [kwɑ'drætɪk /kwɒ-] n. 二次方程式adj. 二次的7.square [skwer /skweə] n.正方形，方块adj. 平方的v. 使…成正方形8.rational ['ræʃənl] adj. 理性的，推理的，合理的9.exponent [ek'spəʊnənt] n. 说明者，说明物10.positive ['pɑzətɪv /'pɒz-] n. 实在的事物，正面，正数adj. 肯定的，绝对的，积极的11.negative ['negətɪv] n. 否定，底片，负数adj. 否定的，负的，消极的12.root [ruːt] n. 根，根部v. 生根，来源于，固定13.cube [kjuːb] n. 立方体，立方v. 使成立方形，量…的体积14.surd [sə:d] n. 无理数，不尽根数15.decimal ['desɪml] n. 小数adj. 十进位的，小数的16.fraction ['frækʃn] n. 分数，破片，小部分17.prime [praɪm] n. 最初，春天，黎明adj. 最初的，基本的adj. 质数的，互为质数的18.rationalize ['ræʃnəlaɪz] v. 使合理化，使有理化Chapter 2 Quadratic functions19.discriminant [dɪ'skrɪmɪneɪt] v. 区别，辨别，有差别地对待20.symmetry ['sɪmɪtrɪ] n. 对称，匀称，调和21.factorization [ˌfæktəraiˈzeiʃən] n. 因数分解22.parabola [pə'ræbələ] n. 抛物线23.coefficient [kəʊɪ'fɪʃnt] n. 系数24.formula ['fɔrmjələ/-mjʊl-] n. 公式，客套语，规则25.appropriate [ə'prəʊprɪət] v. 拨出，挪用adj. 适当的，恰当的，相称的26.sketch [sketʃ] n. 素描，小品，草图v. 为...绘草图，草拟，画素描，画草图27.axis ['æksɪs] n. 轴，主径，轴线bine [kəm'baɪn] n. 集团，企业联合v. 使结合，结合，化合，兼备，兼有29.coordinate [kəʊ'ɔːdɪneɪt] n. 同等物，坐标v. 调整，整理adj. 同等的，并列的30.plot [plɒt] n. 小块土地，图，地区图v. 划分，密谋，画图31.determine [dɪ'tɜːmɪn] v. 决定，判决，裁定，确定，终止32.rearrange [rɪːə'reɪndʒ] v. 再排列，重新整理Chapter 3 Equations and inequalities33.inequality [ɪnɪ'kwɑlətɪ/-'kwɒl-] n. 不均等，不平衡，不平等，不平坦34.sequence ['sɪːkwəns] n. 顺序，序列，继起的事35.series ['sɪrɪːz /'sɪə-] n. 连续，丛书，系列36.elimination [ɪ‚lɪmɪ'neɪʃn] n. 排去，消去，除去37.simultaneous [saɪməl'teɪnɪəs /‚sɪm-] adj. 同时的，同时发生的38.linear ['lɪnɪə(r)] adj. 线的，线状的，直线的39.eliminate [ɪ'lɪmɪneɪt] v. 除去，剔除，排除40.graphically ['græfɪklɪ] adv. 通过图表，轮廓分明地，生动地41.straight [streɪt] n. 直线，直adj. 直的，连续的adv. 直接地，不断地42.intersect [ɪntə(r)'sekt] v. 横断，相交，交叉43.substitution [sʌbstɪ'tuːʃn /-'tjuːʃn] n. 代替，代入法，取代作用44.substitute ['sʌbstɪtuːt /-tjuːt] v. 代替，替代，取代n. 代理，代用品，代理人45.opposite ['ɑpəzɪt /'ɒp-] n. 对立面，对立物adj. 相对的，对立的46.overlap [ˌəuvəˈlæp] n. 重叠，重复v. 重叠，重复，部分相同，部分同时发生47.factorise [ˈfæktəraiz] v. 可分解为因数48.rough [rʌf] n. 粗制品adj. 粗糙的，草率的adv. 粗糙地，粗暴地v. 使不平49.approach [ə'prəʊtʃ] n. 门径，接近v. 靠近，接近50.rectangular [rek'tæŋgjələ(r)] adj. 矩形的，长方形的，直角的，有直角的Chapter 4 Sketching curves51.cubic ['kjuːbɪk] adj. 立方体的，立方的52.reciprocal [rɪ'sɪprəkl] n. 互相作用的事物，倒数adj. 相互的，交互的，对等的53.transformation [trænsfə(r)'meɪʃn] n. 变化，转化54.differentiation [dɪfərenʃɪ'eɪʃn]n. 区别，微分，变异55.technique [tek'nɪːk] n. 技巧，技法，手段，技术，方法plex ['kɒmpleks] n. 复合物，综合体adj. 复杂的，合成的57.identical [aɪ'dentɪkl] adj. 同一的，完全相同的，完全相似的，同源的58.gradient ['greɪdɪənt] n. 倾斜度，升降率，坡度59.reflection [rɪ'flekʃn] n. 反射，反射光，映像，倒影60.intersection [ɪntə(r)'sekʃn] n. 交集，交叉点，十字路口61.interpret [ɪn'tɜrprɪt /-'tɜːp-] v. 解释，诠释，说明，口译，翻译，理解62.correspond [kɒrɪ'spɒnd] v. 符合，通信，协调63.translation [trænz'leɪʃn ,-s-] n. 翻译，转化，译文64.indicate ['ɪndɪkeɪt] v. 指出，象征，显示65.inflexion [inˈflekʃən] n. 变音，转调，弯曲，拐点66.asymptote [ˈæsimptəut] n. 渐近线67.vertical ['vɜrtɪkl /'vɜːt-] adj. 垂直的，直立的n. 垂直线，垂直面68.horizontal [hɑrɪ'zɑntl /‚hɒrɪ'zɒntl] adj. 地平线的，水平的n. 水平线，水平面69.tend [tend] v. 走向，倾向，趋向70.dash [dæʃ] n. 冲撞，破折号v. 泼溅，掺和，使猛撞，冲撞71.quadrant ['kwɑdrənt /'kwɒd-] n. 四分圆，四分仪，象限72.category ['kætɪgərɪ] n. 种类，范畴，别73.diagram ['daɪəgræm] n. 图标v. 用图解法表示，图示74.steep [stɪːp] adj. 陡峭的，急剧升降的v. 泡，浸泡，使埋头75.hence [hens] adv. 因此，从此76.stretch [stretʃ] n. 伸展，绵延，张开v. 伸直，伸长，伸出77.scale [skeɪl] n. 刻度，尺度，比率v. 攀登，向上，调节图片的尺寸78.integer ['ɪntɪʒər /-dʒə] n. 完整的事物，整数，整体Chapter 5 Coordinate geometry in the (x, y) plane79.perpendicular [pɜrpən'dɪkjələr /‚pɜːpən'dɪkjʊlə] adj. 垂直的，直立的n. 垂直线80.tangent ['tændʒənt] n. 切线，正切81.normal ['nɔrml /'nɔːm-] n. 常态，正常，法线adj. 正常的，标准的82.intercept [ɪntə(r)'sept] v. 拦截，截击，截住，截断83.decimal ['desɪml] n. 小数adj. 十进位的，小数的84.parallel ['pærəlel] n. 平行，相匹敌之物v. 与…平行，相比adj. 平行的，相似的85.origin ['ɑrɪdʒɪn ,'ɔ- /'ɒr-] n. 起源，起因，由来，出身86.calculate ['kælkjʊleɪt] v. 计算，预测，估计，推测87.divide [dɪ'vaɪd] n. 分歧，不合v. 分，划分88.numerator ['nuːməreɪtə/'njuː-] n. 分子，计算器，计算者89.denominator [dɪ'nɒmɪneɪtə] n. 分母，命名者90.expand [ɪk'spænd] v. 使膨胀，扩张，张开，发展91.bracket ['brækɪt] n. 支架，托架，括弧v. 括入括弧Chapter 6 Sequences and series92.sequence ['sɪːkwəns] n. 顺序，序列，继起的事93.series ['sɪrɪːz /'sɪə-] n. 连续，丛书，系列94.recurrence [rɪ'kɜrəns /-'kʌrəns] n. 再发生，复发，再现，重新提起，回忆95.arithmetic [ə'rɪθmətɪk] n. 计算，算术，估计adj. 算术的，根据算术的96.symbol ['sɪmbl] n. 符号，象征，记号97.generate ['dʒenəreɪt] v. 产生，导致，发生98.relationship [rɪ'leɪʃnʃɪp] n. 关系，关联，亲属关系，人际关系99.multiple ['mʌltɪpl] n. 倍数，并联adj. 复合的，多样的，并联的100.investor [ɪn'vestə(r)] n. 投资者，出资者101.deposit [dɪ'pɒzɪt] n. 存款，堆积物，定金v. 存放，堆积，沉淀102.reputedly [rɪ'pjuːtɪdlɪ] adv. 据说，根据风评103.equality [iː'kwɒlətɪ] n. 等同性，平等，同等104.salary ['sælərɪ] n. 薪资，薪水，工资105.polygon ['pɑlɪgɑn /'pɒlɪgən] n. 多角形，多边形106.inclusive [ɪn'kluːsɪv] adj. 包含的，包括的mission [com·mis·sion || kə'mɪʃn] n. 佣金，任务，职权，委员会v. 委任108.drill [drɪl] n. 钻，钻头，钻机，钻床v. 钻，钻出109.scheme [skɪːm] n. 方案，图谋，体制v. 计划，策划，密谋110.available [a'vail·a·ble || -ləbl] adj. 有空的，有用的Chapter 7 Differentiation111.differentiation [dɪfərenʃɪ'eɪʃn]n. 区别，微分，变异112.straight [streɪt] n. 直线，直adj. 直的，连续的，正直的adv. 直接地，不断地，立即113.specific [spɪ'sɪfɪk] n. 特性，详情adj. 特殊的，明确的114.estimate ['estɪmeɪt] n. 估计，判断v. 估计，判断，评价115.tangent ['tændʒənt] n. 切线，正切116.inspection [ɪn'spekʃn] n. 检查，视察117.deduce [dɪ'djuːs] v. 推论，演绎出118.chord [kɔːd] n. 和弦，和音，曲线上两点之间的线119.steep [stɪːp] adj. 陡峭的，急剧升降的v. 泡，浸泡，使埋头120.derive [dɪ'raɪv]v. 得自，起源121.definition [defɪ'nɪʃn] n. 定义，清晰度，精确度122.expression [ɪk'spreʃn] n. 表达，表达式，措辞，语法123.power ['paʊə(r)] n. 幂，乘方n. 权，政权，权势v. 使…有力量124.alternative [al'ter·na·tive || -nətɪv] n. 选择，供选择的东西adj. 二者择一的125.polynomial [pɑlɪ'nəʊmɪəl /‚pɒ-] n. 多词学名，多项式126.differentiate [dɪfə'renʃɪeɪt] v. 使有差异，区别，区分，鉴别，求导127.separate ['sepəreɪt] v. 分隔，使分离，分割adj. 单独的，分开的，各别的128.fraction ['frækʃn] n. 分数，破片，小部分129.derivative [dɪ'rɪvətɪv] n. 引出之物，衍生字，系出物adj. 引出的，系出的130.notation [nəʊ'teɪʃn] n. 标记法，乐谱，记号，标志131.particular [pər'tɪkjələr /pə'tɪkjʊlə] n. 个别项目，详细说明adj. 特别的，挑剔的132.displacement [dis'place·ment || -mənt] n. 移置，取代，转移133.temperature ['tempərtʃə(r) /-prə-] n. 温度，热度，发烧134.variable ['verɪəbl /'veər-] n. 变数，可变物adj. 可变的，易变的，不定的135.independent [ɪndɪ'pendənt] n. 中立派，无党派者adj. 独立自主的，不受约束的136.radius ['reɪdɪəs] n. 半径，辐射光线，范围137.represent [reprɪ'zent] v. 描绘，表现，表示，象征，作为…的代表138.velocity [vɪ'lɑsətɪ/-'lɒs-] n. 速度，迅速，速率139.acceleration [æk‚selə'reɪʃn] n. 加速，加速度，促进140.rate [reɪt] n. 比例，率，速度，速率，比率，费用，价格v. 对…估价，认为141.corresponding [cor·re'spond·ing || ‚kɑdɪŋ] adj. 符合的，相同的，一致的142.triangle ['traɪæŋgl] n. 三角，三角板，三角尺143.constant ['kɑnstənt /'kɒn-] n. 常数，恒量adj. 不变的，坚决的，持续的Chapter 8 Integration144.integration [ɪntɪ'greɪʃn] n. 整合，集成，完成145.integrate ['ɪntɪgreɪt] v. 综合，使成整体，使结合146.principle ['prɪnsəpl] n. 原则，主义，原理，信条147.integral ['ɪntɪgrəl] n. 积分，整数adj. 整体的，积分的，整数的148.simplify ['sɪmplɪfaɪ] v. 简化，精简，使平易，单纯149.derivative [dɪ'rɪvətɪv] n. 引出之物，衍生字，系出物adj. 引出的，系出的150.property ['prɑpə(r)tɪ/'prɒ-] n. 财产，性质，所有权151.origin ['ɑrɪdʒɪn ,'ɔ- /'ɒr-] n. 起源，起因，由来，出身。
高等数学双语教材答案
高等数学双语教材答案Chapter 1: Limits and ContinuitySection 1: Introduction to Limits and ContinuityThe concept of limits and continuity is fundamental in higher mathematics. In this section, we will introduce the basic definitions and properties associated with limits and continuity.1.1 Definitions of LimitsIn order to understand limits, we need to define what it means for a function to approach a particular value. Let f(x) be a function defined on an open interval containing c, except possibly at c. We say that the limit of f(x) as x approaches c is L, denoted bylim (x→c) f(x) = L, if for every ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - c| < δ.1.2 Basic Limit LawsOnce we have a clear understanding of limits, we can explore some basic laws that govern their behavior. These laws include the sum law, constant multiple law, product law, quotient law, and the power law.1.3 ContinuityA function f(x) is said to be continuous at a point c if three conditions are met: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c). We can also discuss continuity on an interval or at infinity.Chapter 2: DifferentiationSection 1: Introduction to DifferentiationDifferentiation is an important concept in calculus that allows us to find the rate at which a function is changing at any given point. In this section, we will introduce the concept of differentiation and its applications.2.1 Derivative DefinitionThe derivative of a function f(x) at a point c is defined as the limit of the difference quotient as h approaches 0. Mathematically, this can be written as f'(c) = lim (h→0) [(f(c + h) - f(c))/h].2.2 Differentiation RulesThere are several rules that allow us to find the derivative of a function quickly. These rules include the constant rule, power rule, sum rule, difference rule, product rule, quotient rule, and chain rule.2.3 Applications of DifferentiationDifferentiation has many applications in various fields, such as physics, economics, and engineering. It can be used to find maximum and minimum values, determine rates of change, and solve optimization problems.Chapter 3: IntegrationSection 1: Introduction to IntegrationIntegration is the reverse process of differentiation. It enables us to find the area under a curve and solve various mathematical problems. In this section, we will introduce the concept of integration and its applications.3.1 Indefinite IntegralsThe indefinite integral of a function f(x) is the collection of all antiderivatives of f(x). It is denoted by ∫ f(x) dx and represents a family of functions rather than a single value.3.2 Integration TechniquesThere are various techniques for finding antiderivatives and evaluating definite integrals. These techniques include basic integration rules, substitution, integration by parts, and trigonometric substitution.3.3 Applications of IntegrationIntegration has numerous applications, such as finding the area between two curves, calculating the length of curves, determining volumes of solids, and solving differential equations.ConclusionIn conclusion, the study of high-level mathematics, particularly limits, continuity, differentiation, and integration, is crucial for a comprehensive understanding of advanced mathematical concepts. This article has provided a brief overview of these topics, highlighting their definitions, properties, and applications. By mastering these concepts, students can develop strong problem-solving skills and apply them in various academic and real-world scenarios.。
3rd Edition Chapter 3第三版3章
整理ppt 3-8
Chapter 3 outline
3.1 Transport-layer services
3.2 Multiplexing and demultiplexing
3.3 Connectionless transport: UDP
3.4 Principles of reliable data transfer
整理ppt 3-4
Transport vs. network layer
network layer: logical
communication between hosts
transport layer: logical
communication between processes
relies on, enhances, network layer services
Multiplexing at send host:
gathering data from multiple sockets, enveloping data with header (later used for demultiplexing)
application P3 transport network link physical
UDP: connectionless transport
TCP: connection-oriented transport
TCP congestion control
整理ppt 3-2
Chapter 3 outline
3.1 Transport-layer services
3.2 Multiplexing and demultiplexing
科技英语翻译-第3章
Chapter Three Translation of Nouns & Articles
名词的翻译 一、直译 所谓直译，就是把原文中名词或名词短语本身的含义翻 译出来。如： A power reactor has no need of air, for the heat generated in the uranium pile is the result of unclear fission, not of combustion. 动力反应堆不需要空气，因为在铀堆中产生的热室核裂 变而不是燃烧的结果。 Differentiation can also be expressed in terms of biochemical activities. 分化也可以用生化活动来表达。
Chapter Three Translation of Nouns & Articles
名词的翻译 三、增译 名词的增译多数是为了使译文更符合汉语的表达方式与习惯。如： Breeding of hummingbirds takes? place in May and June. 蜂鸟生育繁殖的时间是每年的五月和六月。 译文在breeding后增译了“时间”;在in May and June前增译了 “每年”，使整个句子读起来更加通顺，意思表达更准确完整。 A drilled hole can be made accurate and smooth by a? reamer. 钻好的孔可用铰刀加工，使其尺寸精确，表面光滑。 该句在形容词accyrate and smooth前分别增译了名词“尺寸”和 “表面”，译文的意思表达的更明确一些。
Chapter Three Translation of Nouns & Articles
chapter-03
Chapter 3. Specialized TerminologiesThe specialized vocabulary used in various scientific disciplines has precise meaning to those engaged in that discipline, but occasionally a different meaning to scientists practic-ing a different discipline.Professional societies try to present information in their journals as clearly as possible to their readers. This manual should be used as a primary source for conventions and style in all ASA, CSSA, SSSA publications. Other style manuals supplement this manual, including Scientific Style and Format (CSE, 2006), the ACS Style Guide (Coghill and Garson, 2006), the Chicago Manual of Style (UCP, 2010), and the US Government Printing Office Style Manual, 2008 (USGPO, 2008). Authors are also encouraged to study recent issues of ASA, CSSA, SSSA journals and books for the general style and format used.Except as new terminology itself forms the content of a paper (as in reports on gene names for a given crop, or proposals for new evaluation scales), authors should avoid making up new terms. If new developments seem to call for new terms, authors should still consult others who normally work in the field in question before trying to devise a new terminology. It is also wise to do a literature search for related materials published by the Societies and elsewhere to see if a consensus on terminology exists or is emerging. In some cases, simply consulting a good dictionary, or the chapters on specialized terms in the major scientific style manuals, is enough to resolve a terminology question.A number of committees of ASA, CSSA, and SSSA have studied terminology in specialized fields and in many cases have indicated a preference.CROP SCIENCE GLOSSARYGlossary of Crop Science Terms is available on the CSSA Website (www.crops. Theorg/publications/crops-glossary).Earlier lists of terms compiled by various committees on crop terminology were pub-lished in Crop Science (Leonard et al., 1968; Shibles, 1976). These reports cite relevant articles and lists published in related fields and include previously published reports issued by earlier committees. In addition, letters in the journal may comment on various aspects of terminology (e.g., Dybing, 1977).SOIL SCIENCE GLOSSARYGlossary of Soil Science Terms is available both in hard copy (SSSA, 2008) and Theon the SSSA Website (/publications/soils-glossary). It contains definitions of more than 1800 terms, a procedural guide for tillage terminology, an outline of the US soil classification system, and the designations for soil horizons and layers. Obsolete terms are noted as such.SPECIALIZED TERMINOLOGYCrop Growth Staging ScalesThe CSSA Ad Hoc Committee on Growth Staging for CSSA Publications (C392.1) in 1996 developed a list of growth staging scales for society publications. The committee recommends that staging scales be used in all ASA, CSSA, SSSA publications when refer-ring to the morphological development stage of plants. References for crop-specific scales recommended by the committee for some major crops are listed in Table 3–1. This list is not intended to include all scales in the literature, but rather the most recent versions for some major crops. If no staging scale exists for a crop, it is recommended that the BBCH (BASF–Bayer–Ciba-Geigy–Hoechst) scale be used (Lancashire et al., 1991).Copyright © ASA–CSSA–SSSA, 5585 Guilford., Madison, WI 53711, USA.Soil IdentificationAll soils discussed in publications of ASA, CSSA, and SSSA should be identified according to the US soil taxonomic system or World Reference Base for Soil Resources the first time each soil is mentioned. Taxonomic identification given in the abstract need not be repeated in the text. If possible, give the series name in addition to the family name. If the series name is not known, give the family name. If the family name is not known, give the subgroup or a higher category name. At a minimum, specify the great group (the one-word name that is the third-highest taxon, beneath suborder and order; e.g., Dystroxerepts, Fragiudalfs, Medisaprists, Natrargids).The descriptive name may be in the singular or plural, according to meaning. Use the singular form if the reference is to a single pedon or polypedon or to a single class.E xamplEs:• The soil material used in this study was collected from the A horizon of a Brookston pedon (a fine-loamy, mixed, mesic Typic Argiaquoll).• A Cisne soil, fine, smectitic, mesic Vertic Albaqualf, was described and sampled at this site.• Criteria for the Typic Hapludult subgroup were examined.• Ontario soils, in the fine-loamy, mixed, mesic Glossoboric Hapludalf family, were studied in greater detail.Use the plural form in reference to several or all of the soils (polypedons) of a class.E xamplEs:• Soils of the Ramona series (fine-loamy, mixed, thermic Typic Haploxeralfs) were treated.• All soils used in the experiments are Typic Dystrochrepts.Table 3–1. Some recommended staging scales and sources for ASA, CSSA, SSSA publications. Recommendations are as developed by the Ad Hoc Committee on Growth Staging for CSSA publications (C392.1) in 1996.Crop CitationAlfalfa Kalu and Fick (1981)Fick and Mueller (1989)†Corn Ritchie et al. (1996)Cool-season forage grasses Haun (1973)Moore et al. (1991)Cotton Elsner et al. (1979)Red clover Ohlsson and Wedin (1989)Small-grain cereals Haun (1973)Zadoks et al. (1974)Tottman (1987)‡Sorghum Vanderlip and Reeves (1972)Soybean Fehr and Caviness (1977)Ritchie et al. (1994)§Stoloniferous grasses West (1990)Sunflower Schneiter and Miller (1981)Warm-season forage grasses Moore et al. (1991)Sanderson (1992)All crops and weeds Lancashire et al. (1991)¶† Enhancement of Kalu and Fick (1981).‡ Enhancement of Zadoks et al. (1974).§ Enhancement of Fehr and Caviness (1977).¶ The BBCH (BASF–Bayer–Ciba-Geigy–Hoechst) scale as presented by Lancashire et al. (1991) can be used for all other crops and weeds.Copyright © ASA–CSSA–SSSA, 5585 Guilford Rd., Madison, WI 53711, USA.For field experiments, the soil present in the plots or fields should be identified, preferably as phases of soil series so that surface texture and slope are known in addition to profile properties. Any dissimilar inclusions that are present also should be named and their extent suggested. It also may be appropriate to name and briefly describe the common soils of the area surrounding the study site. Use the present tense if the soil still exists or reasonably is thought to still exist. E xamplE:The 5-ha study area is mapped as Yolo silt loam, 0 to 2% slopes. The Yolo soils are fine-silty, mixed, nonacid, thermic Typic Xerorthents. Small areas of Cortina very gravelly sandy loam soils (loamy-skeletal, mixed, superactive, nonacid, thermic Typic Xerofluvents) occupy about 10% of the study area.The US taxonomic system should be identified as the US soil taxonomy at first use, after which it may be referred to as Soil Taxonomy. Amendments to Soil Taxonomy (Soil Survey Staff, 1999) have been issued in the National Soil Survey Handbook (http://www. /wps/portal/nrcs/detail/soils/survey/?cid=nrcs142p2_054242) and in Keys to Soil Taxonomy (Soil Survey Staff, 2014). Additional issues of the handbook and new ver-sions of the keys manual can be expected. Updated versions of these and other resources are available online at the Soil Survey home page ().If possible, consult with members of the National Cooperative Soil Survey (NCSS) and check the current USDA-NRCS official soil series descriptions (https:///osdname.aspx) for proper identification of soil designations and nomenclature for soil horizons.For soils outside the United States, authors are encouraged to give soil identifica-tion according to Soil Taxonomy in addition to the identification in their national system.E xamplE:Soil at the site is a Hythe clay loam, classified as a fine, montmorillonitic, frigid Mollic Cryoboralf in the USDA classification (Soil Survey Staff, 1994) and a Gray Luvisol in the Canadian classification (Canada Soil Survey Committee, 1978).Munsell Color NotationMunsell color notations may be used alone in text, tables, or figures. First mention in the abstract or text may be accompanied by the appropriate word descriptions in paren-theses, thus: 10YR 5/4 (yellowish brown).Light Measurements and PhotosynthesisPublications of the ASA, CSSA, and SSSA use the radiometric system with SI units denoting the energy or the quantum content of the radiation used by plants. (See also Chapter 7.)Terms recommended by the Committee on Crop Terminology for the expression of photosynthetic energy and photosynthetic capacity are as defined by Shibles (1976). These terms, with their suggested abbreviations and units, are as follows.1• Photosynthetically active radiation (PAR): radiation in the 400- to 700-nm wave-band.1 Since 1976, the Societies have abandoned the einstein (a name for 1 mole of photons) in favor of the mole. Note that in the original Shibles (1976) article, the typographic errors “nE” and “nmol” are to be read as µE and µmol.Copyright © ASA–CSSA–SSSA, 5585 Guilford., Madison, WI 53711, USA.• Photosynthetic photon flux density (PPFD): the number of photons in the 400- to 700-nm waveband incident per unit time on a unit surface. Suggested units: µmol m−2 s−1.• Photosynthetic irradiance (PI): the radiant energy in the 400- to 700-nm waveband incident per unit time on a unit surface. Suggested units: W-m−2.• Apparent photosynthesis (AP): photosynthesis estimated indirectly and uncor-rected for respiratory activity. The term apparent photosynthesis is preferred to ‘net photosynthesis’ or ‘net assimilation’, because the latter terms imply measurement of a photosynthetic product.• CO2 exchange rate (CER): The net rate of carbon dioxide diffusion from (−) or to (+) an entity, such as a plant tissue, organ or canopy, a soil surface, etc. Suggested units: µmol cm−2 s−1. (Use this term instead of "net CO2 exchange" except in the rare instance when the measurement does not involve a rate.)Reporting PAR in photon units (PPFD) is preferred to energy units (PI), but both are acceptable. Because irradiance is specifically defined in energy units (W), the term cannot be applied to photon flux density.Abandoned as a term is light intensity to denote the amount of light incident on a surface (Dybing, 1977). The Crop Science editorial board has discontinued the use of the photometric system and units scaled to the response of the human eye.SPECIALIZED TERMINOLOGY IN RELATED FIELDSBiologyIdentify all organisms at first mention. For plants, pathogens, and insects and related pests, give both a common name and the scientific name. For plants, include the authority.E xamplE:Sorghum [Sorghum bicolor (L.) Moench] was. . . .The scientific name, also known as the Latin name, is the two-part genus–species bino-mial—or, for subspecies and varieties, the trinomial. For abbreviations of authorities, the primary source is Authors of Plant Names by Brummitt and Powell (1992). If the first mention is in the abstract, the scientific name need not be repeated in the text. Common names, if they exist and are not in dispute, are used in titles of articles, chapters, and books.For the names of crops, use the singular. Although the ordinary English preference is for terms such as oats, beans, and peas, the formal name of a crop defined by a single genus or species is given in the singular: oat, bean, pea, soybean, and so forth. This rule applies even when discussing multiple types of a crop.For common names that are taxonomically inaccurate, join the parts into a single word. For example, writing "pigeonpea" and "chickpea" as one word indicates that these are not Pisum species; similarly, the absence of a space in the common name indicates that Douglasfir is not an Abies species. The USDA-ARS style (set solid) is preferred to the USDA Forest Service style (hyphenated, the traditional usage for forestry).Correct scientific names are in accordance with published rules. For plants, the International Code of Botanical Nomenclature (McNeill et al., 2006; http://ibot.sav. sk/icbn/main.htm) governs; updates appear in Regnum Vegetabile as mandated by the International Botanical Congress, which meets every six years. For cultivated plants, the rules of nomenclature are published as the International Code of Nomenclature for Cultivated Plants (Brickell et al., 2009; /chronica/pdf/sh_10.pdf).Copyright © ASA–CSSA–SSSA, 5585 Guilford Rd., Madison, WI 53711, USA.A practical guide to these codes and to the standards for animals, bacteria, and viruses is published in Scientific Style and Format (CSE, 2006, Chapters 21–24).The scientific names for larger animals (e.g., sheep) do not need to be given unless germane to the article and/or there may be confusion as to what animal is being discussed. Virus species do not have Latin names, but the name of the virus (as approved by the International Committee on Taxonomy of Viruses) should be written in italics, with the first word capitalized (e.g., Tomato spotted wilt virus).To find up-to-date scientific names, consult one of the major online databases: • https:///gringlobal/taxon/taxonomysimple.aspx for plants, especially economic plants (USDA National Plant Germplasm System, Germplasm Resources Information Network [GRIN] database)• for plants, especially noncrop plants (USDA-NRCS) • /fungaldatabases/index.cfm for fungi (USDA Systematic Botany and Mycology Laboratory; Farr and Rossman, 2012)• /publications/commonnames/Pages/default.aspx for plant disease names (American Phytopathological Society)• h ttp:/// for insect scientific names (Texas A&M University) • /index.asp (International Committee on Taxonomy of Viruses) The International Plant Names Index, a product of a collaboration between the Royal Gardens, Kew, the Harvard University Herbaria, and Australian National Herbarium, is available online (/index.html). (This replaces the Kew Index.) Standard printed reference works for nomenclature include Hortus III (Bailey, 1976) and World Economic Plants: A Standard Reference (Wiersema and León, 1999) for plants; Farr et al. (1989) for fungi; Bergey’s manual (Garrity et al., 2001–2011) for bacteria; and, for viruses, Büchen-Osmond (2003).The terms cultivar and variety are synonymous as applied to names of cultivated plants, but cultivar is strongly preferred, to avoid confusing cultivated variety (a term of convenience) with botanical variety (a subtaxon to species).Crop cultivars must be identified as such at first mention in abstract or text. This identification may be given in one of the following two ways:1. By single quotation marks inside punctuation. E xamplE: ‘Vernal’ al f alfa orMedicago sativa L. ‘Vernal’.2. By use of the word cultivar. E xamplE: the cultivar Vernal.Journal of Plant Registrations publishes articles on registered cultivars, germplasms, parental lines, genetic stocks, and mapping populations. Information on these registrations is also available from the GRIN database (https:///gringlobal/search. aspx), usually with some additional narrative. The database entries include pending regis-trations and are linked to plant variety protection status.Citing Genetic MaterialAuthors of CSSA publications must cite plant introductions, as well as registered cul-tivars, germplasm, parental lines, and genetic stocks when they are mentioned in the text of the Introduction, Discussion, or Characteristics section of research papers. Such genetic materials must also be cited when they are used to develop unreleased genetic populationsCopyright © ASA–CSSA–SSSA, 5585 Guilford., Madison, WI 53711, USA.that are the focus of the research paper, unless the development of the population can be cited more directly. Authors are encouraged to cite the Journal of Plant Registrations if possible. Other sources for citation information include GRIN, maintained by the USDA. Registrations published in Crop Science and the Journal of Plant Registrations are indexed on the GRIN Website at https:///gringlobal/query/query.aspx. A gen-eral search in GRIN is available at https:///gringlobal/search.aspx. Reference ExamplesLewis, J.M., L. Siler, E. Souza, P.K.W. Ng, Y. Dong, G. Brown-Guedira, G.-L. Jiang, and R.W. Ward. 2010. Registration of ‘Ambassador’ wheat. J. Plant Reg. 4:195–204. USDA-ARS National Genetic Resources Program. 1993. Germplasm Resources Information Network (GRIN) database. Festuca arundinacea Schreb. POACEAE ‘Maximize’. National Germplasm Resources Laboratory, Beltsville, MD. http:// /cgi-bin/npgs/acc/display.pl?1444051 (accessed 1 Feb. 2012).Genetics and Molecular and Cell BiologyGenes are named according to established conventions, which vary in part among crops. As an example, a standard for cotton is Kohel (1973). Many of these are summa-rized in Scientific Style and Format (CSE, 2006, p. 298–312); see also the entries for gene and genotype in the New Oxford Dictionary for Scientific Writers and Editors (Martin, 2009). Check with an expert in your field to find the appropriate published standards, including updates. Accepted names of genes are set in italics and may be modified with letters or numbers (with or without superscripts, with or without italics). Proposed names follow the conventions for the crop in question but are set in roman type.Use italics for the variables in ploidy formulas (e.g., 2n = 2x = 42).Spell out amino acids in text, without capitalization. In formulas and sequences, use the abbreviations shown in Table 3–2.For enzymes, follow nomenclature for name and number (Webb, 1992; http://www. /iubmb/enzyme/).As for genetics, the CSE manual (CSE, 2006) is an excellent guide to style for spe-cialized terms and usages in molecular and cell biology, as is the New Oxford Dictionary for Scientific Writers and Editors (Martin, 2009). The Oxford book gives, for example, complete rules for names of restriction enzymes: three letters in italics to identify the source bacterium (e.g., Hin for Haemophilus influenzae, or Bam for Bacillus amylolique-faciens), then letters in roman type to indicate the strain (e.g., d or H), then capital roman numerals to indicate the type of enzyme (e.g., I, II, or III), all leading to characteristic names such as Hin dIII (for enzyme III from strain d of H. influenzae) or Bam HI (for enzyme I from strain H of B. amyloliquefaciens).ChemistryUse chemical symbols instead of words for elements, ions, or compounds, except at the beginning of a sentence. These symbols do not have to be defined the first time they are used. Where the representation is general and the chemical species is not specified, do not indicate the ionic charge (e.g., Ca, Fe, K, NH4, NO3, SO4, and PO4). Whenever a specific ion of known valence state is described in a manuscript, indicate the charge in superscripts as the charge number followed by a plus (+) or minus (−) sign; where the charge number is 1, use only the sign (e.g., Ca2+, NH4+, NO3−). Where the oxidation state is not obvious inCopyright © ASA–CSSA–SSSA, 5585 Guilford Rd., Madison, WI 53711, USA.a formula or where the oxidation state is known and is important, it should be designated by a roman numeral in parentheses; for example, Fe(II).The amounts and proportions of fertilizer nutrient elements must be expressed in terms of the elements or in other ways as needed for theoretical purposes. The amounts or proportions of the oxide forms (P2O5, K2O, etc.) may also be included, in parentheses.Give the full chemical names for compounds at first mention in the abstract or text. (If many names need mention, they may be listed in a table instead of parenthetically throughout the text.) E xamplEs:atrazine [6-chloro-N-ethyl-N′-(1-methylethyl)-1,3,5-triazine-2,4-diamine]cyanazine {2-[[4-chloro-6-(ethylamino)-1,3,5-triazin-2-yl] amino]-2-methyl p ro-panenitrile}If given in the abstract, the full chemical names do not need to be repeated in the text. Use the most up to date chemical names available. Thereafter, the common or generic name can be used (e.g., atrazine, 2,4-D, etc.). Trade names should be avoided whenever possible. If it is necessary to use a trade name, it should be capitalized and spelled out as specified by the trademark owner. Omit the various trademark symbols, such as ® and ™.In the United States and Canada, the authority for names of chemical compounds is Chemical Abstracts and its indexes. The American Chemical Society’s ACS Style Guide (Coghill and Garson, 2006) and the Council of Science Editors’ Scientific Style and Format (CSE, 2006) contain many additional details on nomenclature in chemistry and biochemistry. Publications of the American Chemical Society’s committee on nomencla-ture and the nomenclature commissions of the International Union of Pure and Applied Chemistry (IUPAC) are available through Chemical Abstracts Service, Columbus, OH.Chapter 7 of this book has further information regarding SI units and concentration.Information on pesticides and adjuvants is found in the Herbicide Handbook of the Weed Science Society of America (Ahrens, 1994), the Crop Protection Handbook (Meister,Table 3–2. Amino acids and their abbreviations.Amino acid Long abbreviation Short abbreviation Alanine Ala AArginine Arg R Asparagine Asn NAspartic acid Asp DCysteine Cys CGlutamic acid Glu E Glutamine Gln QGlycine Gly GHistidine His H Isoleucine Ile ILeucine Leu LLysine Lys K Methionine Met M Phenylalanine Phe FProline Pro PSerine Ser S Threonine Thr T Tryptophan Trp W Tyrosine Tyr YValine Val VCopyright © ASA–CSSA–SSSA, 5585 Guilford., Madison, WI 53711, USA.current edition), and the British Crop Protection Society’s Pesticide Manual: A Worldwide Compendium (Tomlin, 2011). See also the Merck Index (O'Neil, 2006, or current edition).The chemical names of the organic substances used for pesticides may include locants and descriptors consisting of numerals, letters (italic, roman, small-capital, or Greek letters), symbols, and words in various combinations. Below is a selection of com-mon usages:• Use italics for the prefixes anti, asym, c, cis, cyclo, d, endo, exo, l, m, n, o, p, r, s, sec, t, tert, and trans. Do not capitalize these prefixes, even at the beginning of a sentence or in a title.• Use italics for the capitalized prefixes R, R*, S, S*, E, and Z and enclose them in parentheses.• Use italics for symbols of chemical elements indicating ligation or attachment to an atom (e.g., O, P, N, S) or when indicating added hydrogen (H).• Use Greek letters to denote position or stereochemistry (e.g., a-amino acids). • Enclose the stereochemistry prefixes for plus and minus in parentheses: (+), (−), and (±).• Use roman (plain) type for multiplying prefixes (e.g., hemi, mono, di, tri, deca;semi, uni, sesqui, bi, ter, deci; bis, tris, decakis).For a full treatment with examples, including details of punctuation and capitalization in various contexts, see the ACS Style Guide (Coghill and Garson, 2006, Chapter 12).Copyright © ASA–CSSA–SSSA, 5585 Guilford Rd., Madison, WI 53711, USA.。
高等数学教材答案下册英语
高等数学教材答案下册英语Unit 1: Functions and Their GraphsChapter 1: Linear Functions1.1 Functions and Their Representations1.2 Domain and Range1.3 Linear Functions and EquationsChapter 2: Quadratic Functions2.1 Graphs of Quadratic Functions2.2 Solving Quadratic Equations2.3 Quadratic Functions and Their Transformations Chapter 3: Exponential and Logarithmic Functions3.1 Exponential Functions and Their Graphs3.2 Logarithmic Functions and Their Graphs3.3 Exponential and Logarithmic EquationsUnit 2: Limits and ContinuityChapter 4: Limits and Continuity4.1 Limits and Their Properties4.2 Continuity and Its Properties4.3 Computing LimitsChapter 5: Derivatives5.1 The Derivative and its Interpretation5.2 Differentiation Techniques5.3 Applications of DerivativesChapter 6: Higher-Order Derivatives6.1 Higher-Order Derivatives and Their Interpretations 6.2 The Chain Rule6.3 Implicit DifferentiationUnit 3: IntegrationChapter 7: Antiderivatives and Indefinite Integrals 7.1 Antiderivatives and Their Properties7.2 Indefinite Integrals7.3 Substitution MethodChapter 8: Definite Integrals and Their Applications 8.1 Definite Integrals and Their Properties8.2 Applications of Definite Integrals8.3 Numerical IntegrationChapter 9: Techniques of Integration9.1 Integration by Parts9.2 Trigonometric Integrals9.3 Trigonometric SubstitutionUnit 4: Differential Equations and Applications Chapter 10: First-Order Differential Equations 10.1 Separable Differential Equations10.2 Linear Differential Equations10.3 Applications of Differential Equations Chapter 11: Applications of Differential Calculus 11.1 Optimization11.2 Related Rates11.3 Newton's MethodChapter 12: Sequences and Series12.1 Sequences and Their Limits12.2 Infinite Series12.3 Convergence TestsUnit 5: Multivariable CalculusChapter 13: Functions of Several Variables 13.1 Functions of Two or More Variables13.2 Partial Derivatives13.3 Optimization of Functions of Two VariablesChapter 14: Multiple Integrals14.1 Double Integrals14.2 Triple Integrals14.3 Applications of Multiple IntegralsChapter 15: Vector Calculus15.1 Vector Fields15.2 Line Integrals15.3 Green's TheoremChapter 16: Differential Calculus of Vector Fields16.1 Gradient Fields and Potential Functions16.2 Divergence and Curl16.3 Stokes' TheoremI hope the above chapters and sections provide a comprehensive overview of the answers to the exercises and problems in the textbook. Remember to utilize this answer key as a useful tool to check your understanding and progress in studying advanced mathematics.。
大一高等数学a教材答案详解
大一高等数学a教材答案详解Chapter 1: Functions and Limits1.1 Introduction to FunctionsIn this chapter, we will explore the concept of functions and their properties. A function is a rule that assigns each element from one set to another set. It is represented by f(x), where x is an element from the domain and f(x) is the output value. Functions can be represented graphically, algebraically, or numerically.1.2 Limits and ContinuityLimits are used to describe the behavior of a function as x approaches a certain value. The limit of a function f(x) as x approaches a can be denoted as limₓ→a f(x). Continuity of a function is determined by the existence of a limit at a certain point and the value of the function at that point.1.3 DifferentiationDifferentiation is the process of finding the derivative of a function. The derivative represents the rate of change of a function at a particular point. It is denoted as f'(x) or dy/dx. The derivative can be used to find the slope of a tangent line, determine critical points, and analyze the behavior of functions.Chapter 2: Derivatives2.1 Basic Rules of DifferentiationIn this chapter, we will discuss the basic rules of differentiation. These rules include the power rule, product rule, quotient rule, and chain rule.These rules allow us to find the derivative of various functions by applying specific formulas and techniques.2.2 Applications of DerivativesDerivatives have various applications in real-life situations. They can be used to find maximum and minimum values, solve optimization problems, determine velocity and acceleration, and analyze growth and decay models. This chapter will address these applications and provide practical examples.2.3 Higher Order DerivativesHigher order derivatives refer to derivatives of derivatives. The second derivative represents the rate of change of the first derivative, while the third derivative represents the rate of change of the second derivative, and so on. Higher order derivatives can provide information about the curvature and concavity of a function.Chapter 3: Integration3.1 Antiderivatives and Indefinite IntegralsAntiderivatives are the opposite of derivatives. They represent the original function whose derivative is equal to a given function. The process of finding antiderivatives is called integration. The indefinite integral represents a family of functions, with the constant of integration accounting for the infinite number of antiderivatives.3.2 Definite IntegralsDefinite integrals are used to calculate the accumulated change of a function over a specific interval. The definite integral of a function f(x) froma tob is denoted as ∫[a, b] f(x) dx. It represents the area under the curve of the function between the limits a and b. This chapter will discuss the properties and techniques of definite integration.3.3 Applications of IntegrationIntegration has various applications, including calculating areas and volumes, solving differential equations, determining average values, and analyzing accumulation problems. These applications will be explored in this chapter, along with practical examples.Chapter 4: Techniques of Integration4.1 Integration by SubstitutionIntegration by substitution is a technique used to simplify integrals by replacing variables or functions. It involves choosing an appropriate substitution and applying the chain rule in reverse. This method can be used to solve complex integrals and make them more manageable.4.2 Integration by PartsIntegration by parts is another integration technique that allows us to find the integral of a product of two functions. It involves choosing one function to differentiate and the other function to integrate. This method is useful for integrating products of functions such as polynomials, exponentials, logarithms, and trigonometric functions.4.3 Trigonometric IntegralsTrigonometric integrals involve integrating functions that contain trigonometric functions like sine, cosine, tangent, secant, etc. These integralscan be solved using trigonometric identities and substitution techniques specific to trigonometric functions.In conclusion, the first-year high school mathematics A textbook provides a comprehensive introduction to functions, limits, derivatives, and integration. It covers the fundamental concepts and techniques necessary for further study in advanced mathematics. By understanding and applying the principles discussed in this textbook, students will acquire a solid foundation in calculus and its applications.。
高等数学（微积分学）专业术语名词概念定理等英汉对照
高等数学（微积分学）专业术语名词概念定理等英汉对照目录第一部分英汉微积分词汇Part 1 English-Chinese Calculus Vocabulary第一章函数与极限Chapter 1 function and Limi t (1)第二章导数与微分Chapter 2 Derivative and Differential (2)第三章微分中值定理Chapter 3 Mean Value theorem of differentials and the Application of Derivatives (3)第四章不定积分Chapter 4 Indefinite Intergrals (3)第五章定积分Chapter 5 Definite Integral (3)第六章定积分的应用Chapter 6 Application of the Definite Integrals (4)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geomertry and Vector Algebra (4) 第八章多元函数微分法及其应用Chapter 8 Differentiation of functions Several variables and Its Application (5)第九章重积分Multiple Integrals (6)第十章曲线积分与曲面积分Chapter 10 Line(Curve ) Integrals and Surface Integral s (6) 第十一章无穷级数Chapter 11 Infinite Series (6)第十二章微分方程Chapter 12 Differential Equation (7)第二部分定理定义公式的英文表达Part 2 English Expression for Theorem,Definition and Formula第一章函数与极限Chapter 1 Function and Limi t (19)1.1映射与函数(Mapping and Function ) (19)1.2数列的极限(Limit of the Sequence of Number) (20)1.3函数的极限(Limit of Function) (21)1.4无穷小与无穷大(Infinitesimal and Inifinity) (23)1.5极限运算法则(Operation Rule of Limit) (24)1.6极限存在准则两个重要的极限(Rule for theExistence of Limits Two Important Limits) (25)1.7无穷小的比较(The Comparison of infinitesimal) (26)1.8函数的连续性与间断点(Continuity of FunctionAnd Discontinuity Points) (28)1.9连续函数的运酸与初等函数的连续性(OperationOf Continuous Functions and Continuity ofElementary Functions) (28)1.10闭区间上联系汗水的性质(Properties ofContinuous Functions on a Closed Interval) (30)第二章导数与数分Chapter2 Derivative and Differential (31)2.1 导数的概念(The Concept of Derivative) (31)2.2 函数的求导法则(Rules for Finding Derivatives) (33)2.3 高阶导数(Higher-order Derivatives) (34)2.4 隐函数及由参数方程所确定的函数的导数相关变化率(Derivatives ofImplicit Functions and Functions Determined by Parametric Equation andCorrelative Change Rate) (34)2.5 函数的微分(Differential of a Function) (35)第三章微分中值定理与导数的应用Chapter 3 Mean Value Theorem of Differentials and theApplication of Derivatives (36)3.1 微分中值定理(The Mean Value Theorem) (36)3.2 洛必达法则(L’Hopital’s Rule) (38)3.3 泰勒公式(Taylor’s Formula) (41)3.4 函数的单调性和曲线的凹凸性(Monotonicityof Functions and Concavity of Curves) (43)3.5 函数的极值与最大最小值(Extrema, Maximaand Minima of Functions) (46)3.6 函数图形的描绘(Graphing Functions) (49)3.7 曲率(Curvature) (50)3.8 方程的近似解(Solving Equation Numerically) (53)第四章不定积分Chapter 4Indefinite Integrals (54)4.1 不定积分的概念与性质(The Concept andProperties of Indefinite Integrals) (54)4.2 换元积分法(Substitution Rule for Indefinite Integrals) (56)4.3 分部积分法(Integration by Parts) (57)4.4 有理函数的积分(Integration of Rational Functions) (58)第五章定积分Chapter 5 Definite Integrals (61)5.1 定积分的概念和性质(Concept of Definite Integraland its Properties) (61)5.2 微积分基本定理(Fundamental Theorem of Calculus) (67)5.3 定积分的换元法和分部积分法(Integration by Substitution andDefinite Integrals by Parts) (69)5.4 反常积分(Improper Integrals) (70)第六章定积分的应用Chapter 6 Applications of the Definite Integrals (75)6.1 定积分的元素法(The Element Method of Definite Integra (75)6.2 定积分在几何学上的应用(Applications of the DefiniteIntegrals to Geometry) (76)6.3 定积分在物理学上的应用(Applications of the DefiniteIntegrals to Physics) (79)第七章空间解析几何与向量代数Chapter 7 Space Analytic Geometry and Vector Algebar (80)7.1 向量及其线性运算(Vector and Its Linear Operation) (80)7.2 数量积向量积(Dot Product and Cross Product) (86)7.3 曲面及其方程(Surface and Its Equation) (89)7.4 空间曲线及其方程(The Curve in Three-space and Its Equation (91)7.5 平面及其方程（Plane in Space and Its Equation) (93)7.6 空间直线及其方程（Lines in and Their Equations) (95)第八章多元函数微分法及其应用Chapter 8 Differentiation of Functions of SeveralVariables and Its Application (99)8.1 多元函数的基本概念（The Basic Concepts of Functionsof Several Variables) (99)8.2 偏导数(Partial Derivative) (102)8.3 全微分(Total Differential) (103)8.4 链式法则（The Chain Rule) (104)8.5 隐函数的求导公式(Derivative Formula for Implicit Functions). (104)8.6 多元函数微分学的几何应用(Geometric Applications of Differentiationof Ffunctions of Severalvariables) (106)8.7方向导数与梯度(Directional Derivatives and Gradients) (107)8.8多元函数的极值(Extreme Value of Functions of Several Variables) (108)第九章重积分Chapter 9 Multiple Integrals (111)9.1二重积分的概念与性质(The Concept of Double Integralsand Its Properities) (111)9.2二重积分的计算法(Evaluation of double Integrals) (114)9.3三重积分(Triple Integrals) (115)9.4重积分的应用(Applications of Multiple Itegrals) (120)第十章曲线积分与曲面积分Chapte 10 Line Integrals and Surface Integrals (121)10.1 对弧长的曲线积分(line Intergrals with Respect to Arc Length) (121)10.2 对坐标的曲线积分(Line Integrals with respect toCoordinate Variables) (123)10.3 格林公式及其应用(Green's Formula and Its Applications) (124)10.4 对面积的曲面积分(Surface Integrals with Respect to Aarea) (126)10.5 对坐标的曲面积分(Surface Integrals with Respect toCoordinate Variables) (128)10.6 高斯公式通量与散度(Gauss's Formula Flux and Divirgence) (130)10.7 斯托克斯公式环流量与旋度(Stokes's Formula Circulationand Rotation) (131)第十一章无穷级数Chapter 11 Infinite Series (133)11.1 常数项级数的概念与性质(The concept and Properties ofThe Constant series) (133)11.2 常数项级数的审敛法(Test for Convergence of the Constant Series) (137)11.3 幂级数(power Series). (143)11.4 函数展开成幂级数(Represent the Function as Power Series) (148)11.5 函数的幂级数展开式的应用(the Appliacation of the Power Seriesrepresentation of a Function) (148)11.6 函数项级数的一致收敛性及一致收敛级数的基本性质(The UnanimousConvergence of the Series of Functions and Its properties) (149)11.7 傅立叶级数（Fourier Series） (152)11.8 一般周期函数的傅立叶级数(Fourier Series of Periodic Functions) (153)第十二章微分方程Chapter 12 Differential Equation (155)12.1微分方程的基本概念（The Concept of DifferentialEquation) (155)12.2可分离变量的微分方程(Separable Differential Equation) (156)12.3齐次方程(Homogeneous Equation) (156)12.4 一次线性微分方程(Linear Differential Equation of theFirst Order) (157)12.5全微分方程(Total Differential Equation) (158)12.6可降阶的高阶微分方程(Higher-order DifferentialEquation Turned to Lower-order DifferentialEquation) (159)12.7高阶线性微分方程(Linear Differential Equation of HigherOrder) (159)12.8常系数齐次线性微分方程(Homogeneous LinearDifferential Equation with Constant Coefficient) (163)12.9常系数非齐次线性微分方程(Non HomogeneousDifferential Equation with Constant Coefficient) (164)12.10 欧拉方程(Euler Equation) (164)12.11 微分方程的幂级数解法(Power Series Solutionto Differential Equation) (164)第三部分常用数学符号的英文表达Part 3 English Expression of the Mathematical Symbol in Common Use第一部分英汉微积分词汇Part1 English-Chinese Calculus V ocabulary 第一章函数与极限Chapter1 Function and Limit集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood领域的半径radius of a neighborhood左领域left neighborhood右领域right neighborhood 映射mappingX到Y的映射mapping of X ontoY 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function 单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于 a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数()f x当x趋于x0时的极限limit of functions () f x as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order 等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数()f x在x0连续the function ()f x is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数()f x在该区间上连续function ()f x is continuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function 导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives()f x在闭区间【a,b】上可导()f x is derivable on the closed interval [a,b]切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error 绝对误差absolute error 相对误差relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives 罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule0/0型不定式indeterminate form of type 0/0不定式indeterminate form泰勒中值定理Taylor’s mean value theorem泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term 麦克劳林公式Maclaurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, cancave up凹向下的，向上凸的concave downward’concave down拐点inflection point函数的极值extremum of function极大值local(relative) maximum最大值global(absolute) mximum极小值local(relative) minimum最小值global(absolute) minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function(antiderivative) 积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an integvals牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution 递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素element of area平面图形的面积area of a luane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of acurve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector 三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积scalar product,dot product,inner product 曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface 椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve 空间曲线的参数方程parametric equations of a space curve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes 点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative 混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function 曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient 数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values 条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test 交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series 阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system 傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation 常微分方程ordinary differential equation偏微分方程partial differential equation,PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation 初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order 自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation 串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency 简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficient第二部分定理定义公式的英文表达Part2 English Expression for Theorem, Definition and Formula第一章函数与极限Chapter 1 Function and Limit1.1 映射与函数 (Mapping and Function)一、集合 (Set)二、映射 (Mapping)映射概念 (The Concept of Mapping) 设X , Y 是两个非空集合 , 如果存在一个法则f ，使得对X 中每个元素x ，按法则f ，在Y 中有唯一确定的元素y 与之对应 , 则称f 为从X 到 Y 的映射 , 记作:f X Y →。
高等数学英文教材书名
高等数学英文教材书名Title: Advanced Calculus - An English TextbookIntroduction:Advanced Calculus is a comprehensive English textbook designed to provide a solid foundation in the principles and applications of higher-level mathematics. This textbook aims to cater to the needs of undergraduate and graduate students studying mathematics, engineering, and other related fields. Its well-structured content, precise explanations, and engaging examples make it an indispensable resource for anyone seeking to deepen their understanding of advanced calculus.Chapter 1: Limits and ContinuityIn this chapter, readers are introduced to the fundamental concepts of limits and continuity. Starting with an overview of the concept of a limit, the textbook gradually delves into the intricacies of one-sided and infinite limits. Through clear explanations and illustrative examples, students will develop a solid understanding of continuity and its various properties, such as the intermediate value theorem and the extreme value theorem.Chapter 2: DifferentiationThis chapter explores the concept of differentiation - a crucial tool in calculus. The textbook provides a step-by-step approach to calculating derivatives of various functions, including polynomials, exponential and logarithmic functions, and trigonometric functions. Additionally, the concept of higher-order derivatives and their applications are extensively covered,allowing students to grasp the intricacies of curve sketching and optimization problems.Chapter 3: IntegrationThe textbook presents integration as a powerful technique for finding areas, volumes, and cumulative sums. Starting with the definite integral and its properties, the chapter progresses to explore various integration methods, such as substitution, integration by parts, and partial fractions. Furthermore, the textbook covers improper integrals and numerical integration methods, equipping students with the necessary tools to solve a wide range of mathematical problems.Chapter 4: Sequences and SeriesThis chapter focuses on the concepts of sequences and series, which play a fundamental role in advanced calculus. Beginning with an introduction to arithmetic and geometric sequences, the textbook moves on to explore the convergence and divergence of sequences. Furthermore, different types of series, including geometric, Taylor, and Fourier series, are presented, along with their convergence properties and applications.Chapter 5: Multivariable CalculusIn this chapter, readers are introduced to the complexities of multivariable calculus. The textbook covers topics such as partial derivatives, directional derivatives, multiple integrals, and vector calculus. Through practical examples and applications, students will develop a deep understanding of how calculus applies to functions of multiple variables, preparing them for more advanced mathematical concepts.Chapter 6: Differential EquationsThe final chapter of this textbook focuses on differential equations, which are essential in modeling real-world phenomena. Starting with first-order differential equations, the textbook progressively introduces higher-order differential equations, separable equations, and systems of linear differential equations. Different solution methods, such as integrating factors and Laplace transforms, are also presented, allowing students to solve a wide range of differential equation problems.Conclusion:Advanced Calculus - An English Textbook provides a comprehensive and rigorous study of advanced mathematical concepts, serving as an invaluable resource for students pursuing mathematics and related disciplines. With its clear explanations, numerous examples, and well-organized content, this textbook equips students with the necessary tools to tackle complex mathematical problems and fosters a deeper understanding of the principles underlying advanced calculus.。
高等数学的英文教材
高等数学的英文教材Higher Mathematics English TextbookIntroduction:Higher Mathematics is a fundamental subject in the field of mathematics, which is extensively studied in higher education institutions worldwide. This English textbook aims to provide a comprehensive guide to higher mathematics concepts, theories, and applications.Chapter 1: Real Numbers1.1 Number Systems- Natural Numbers- Integers- Rational Numbers- Irrational Numbers- Real Numbers1.2 Sets and Functions- Set Theory- Functions and their Types- Mapping and Compositions- Inverse FunctionsChapter 2: Limits and Continuity2.1 Definitions and Properties- The Concept of Limits- Limit Laws and Basic Operations- One-Sided Limits- Infinite Limits2.2 Continuity- Definition and Types of Continuity- Intermediate Value Theorem- Discontinuities and Their Classification Chapter 3: Differentiation3.1 Derivatives- Definition and Notation- Rules of Differentiation- Higher Order Derivatives- Implicit Differentiation3.2 Applications of Differentiation- Tangent and Normal Lines- Optimization Problems- Related Rates- Linear ApproximationChapter 4: Integration4.1 Definite Integrals- Riemann Sums- Fundamental Theorem of Calculus- Techniques of Integration- Improper Integrals4.2 Applications of Integration- Area and Volume- Arc Length and Surface Area- Differential Equations- Applications in Physics and Engineering Chapter 5: Sequences and Series5.1 Sequences- Definitions and Notation- Convergence and Divergence- Arithmetic and Geometric Sequences- Limit and Ratio Tests5.2 Series- Types of Series- Convergence Tests- Power Series- Taylor SeriesChapter 6: Differential Equations6.1 First-Order Differential Equations- Separable Equations- Exact Equations- Linear Equations- Bernoulli Equations6.2 Second-Order Linear Differential Equations - Homogeneous Equations- Non-Homogeneous Equations- Boundary Value Problems- Method of Undetermined Coefficients Chapter 7: Multivariable Calculus7.1 Functions of Several Variables- Domain and Range- Limits and Continuity- Partial Derivatives and Gradients- Maximum and Minimum Values7.2 Multiple Integrals- Double and Triple Integrals- Change of Variables- Applications in 3D Space- Surface and Volume IntegralsConclusion:This Higher Mathematics English Textbook provides a structured and comprehensive overview of various concepts and principles in higher mathematics. With its clear explanations, examples, and applications, it aims to enhance students' understanding and problem-solving abilities in this critical subject area.。
CIE数学学习计划
CIE数学学习计划CIE 数学P1 part:章节主要内容计划课时Chapter1Coordinates, points and lines.1h Chapter2Surds and indices.1hChapter3Functions and graphs.1hChapter4Quadratics .1hChapter5Inequalities .1hChapter6Differentiation .1hChapter7Applications of differentiation.1h Chapter8Sequences.1hChapter9The binomial theorem.1hChapter10Trigonometry.1hChapter11Combining and inverting functions.1h Chapter12Extending differentiation.1hChapter13Vectors.1hChapter14Geometric sequences.1hChapter15Second derivatives .1hChapter16Integration.1hChapter17Volume of revolution1hChapter18Radians1hpast paper真题讲解如果是即将参加考试，讲解past paper时间约分配为8 hour8h 课时合计学生科根据具体章节选择上课内容26hCIE 数学P2 part:章节主要内容计划课时Chapter 1Algebra.1hChapter 2Logarithmic and exponential functions.1hChapter 3Trigonometry.1hChapter 4Differentiation .1hChapter 5Integration .1hChapter 6Inequalities .1hChapter 7Numerical solution of equations.1hpast paper真题讲解如果是即将参加考试，讲解past paper时间约分配为8 hour8h 课时合计学生科根据具体章节选择上课内容15hCIE 数学P3 part:章节主要内容计划课时Chapter 1Algebra.2hChapter 2Logarithmic and exponential functions.2hChapter 3Trigonometry.2hChapter 4Differentiation .2hChapter 5Integration .2hChapter 6Numerical solution of equations .2hChapter 7Vectors.2hChapter 8Differential equations2hChapter 9Complex numbers2hpast paper真题讲解如果是即将参加考试，讲解past paper时间约分配为8 hour8h 课时合计学生科根据具体章节选择上课内容26hCIE 数学M1 part:章节主要内容计划课时Chapter 1Forces and equilibrium.2hChapter 2Kinematics of motion in a straight line.2hChapter 3Newton’s laws of motion.2hChapter 4Energy .2hChapter 5Work and power .2hpast paper真题讲解如果是即将参加考试，讲解past paper时间约分配为6 hour6h 课时合计学生科根据具体章节选择上课内容16hCIE 数学S1 part:章节主要内容计划课时Chapter 1Representation of data.2hChapter 2Permutations and combinations.2hChapter 3Probability.2hChapter 4Discrete random variables .2hChapter 5The normal distribution.2hpast paper真题讲解如果是即将参加考试，讲解past paper时间约分配为6 hour6h 课时合计学生科根据具体章节选择上课内容16h。
Math Proficiency
Chapter 1: Functions and Models (4 lectures)1.1 Four Ways to Represent a Function1.2 Mathematical Models: A Catalog of Essential Functions1.3 New Functions from Old Functions1.5 Exponential Functions1.6 Inverse Functions and LogarithmsChapter 2: Limits and Derivatives (5 lectures)2.1 The Tangent and Velocity Problems2.2 The Limit of a Function2.3 Calculating Limits Using the Limit Laws2.4 The Precise Definition of a Limit (optional)2.5 Continuity2.6 Limits at Infinity; Horizontal Asymptotes 水平渐近线2.7 Derivatives and Rates of Change2.8 The Derivative as a FunctionChapter 3: Differentiation Rules (8 lectures)3.1 Derivatives of Polynomials 多项式and Exponential Functions 3.2 The Product 积and Quotient 商Rules3.3 Derivatives of the Trigonometric Functions3.4 The Chain Rule 链式法则3.5 Implicit Differentiation 隐函数微分3.6 Derivatives of Logarithmic Functions3.7 Rates of Change in the Natural and Social Sciences 3.8 Exponential Growth and Decay3.9 Related Rates3.10 Linear Approximations and Differentials3.11 Hyperbolic Functions 双曲函数Chapter 4: Applications of Differentiation (7 lectures) 4.1 Maximum and Minimum Values4.2 Mean Value Theorem 中值定理4.3 How Derivatives Affect the Shape of a Graph4.4 Indeterminate Forms and L'Hospital's Rule4.5 Summary of Curve Sketching4.7 Optimization Problems4.8 Newton's Method4.9 Antiderivatives不定积分Chapter 5: Integrals (7 lectures)5.1 Areas and Distances5.2 The Definite Integral5.3 The Fundamental Theorem of Calculus5.4 Indefinite Integrals and the Net Change Theorem 5.5 The Substitution RuleChapter 6: Applications of Integration (5 lectures) 6.1 Areas Between Curves6.2 Volumes6.3 Volumes by Cylindrical Shells圆柱体6.4 Work6.5 Average Value of a Function。
高等数学专业英语名词
高等数学术语英语翻译V、X、Z:Value of function：函数值Vector：函数值Volume：体积X-axis：x轴x-coordinate：x坐标x-intercept：x截距Zero vector：函数的零点T:Tangent function：正切函数Tangent line：切线Total differential：全微分Trigonometric function：三角函数Tripe integrals：三重积分S:Second derivative：二阶导数Second partial derivative：二阶偏导数Sequence：数列Set：集合Slope：斜率Smooth curve：平滑曲线Smooth surface：平滑曲面Solid of revolution：旋转体Space：空间Speed：速率Spherical coordinates：球面坐标Sum：和Surface：曲面Surface integral：面积分Surface of revolution：旋转曲面Symmetry：对称Sine function：正弦函数Slant asymptote：斜渐近线R:Range of a function：函数的值域Rate of change：变化率Rational function：有理函数Rational number：有理数Real number：实数Rectangular coordinates：直角坐标Revolution,solid of：旋转体Revolution,surface of：旋转曲面Root：根P、Q:Parabola：拋物线Parabolic cylinder：抛物柱面Paraboloid：抛物面Parallelepiped：平行六面体Parallel lines：并行线Parameter：参数Partial derivative：偏导数Partial differential equation：偏微分方程Partial fractions：部分分式Partial integration：部分积分Partiton：分割Period：周期Periodic function：周期函数Perpendicular lines：垂直线Plane：平面Polar coordinate：极坐标Pole：极点Polynomial：多项式Positive angle：正角Power function：幂函数Product：积M、N、O:Maximum and minimum values：极大与极小值Multiple integrals：重积分Natural num ber：自然数Normal line：法线Number：数Odd function：奇函数One-sided li mit：单边极限Open interval：开区间Ordinary differential equation：常微分方程Orthogonal：正交的Origin：原点L:Law of Cosines：余弦定理Left-hand derivative：左导数Left-hand limit：左极限Length：长度Limit：极限Linear approximation：线性近似Linear equation：线性方程式Linear function：线性函数Linearity：线性Logarithm：对数Logarithmic function：对数函数I:Implicit function：隐函数Increment：增量Indefinite integral：不定积分Independent variable：自变数Indeterminate from：不定型Inequality：不等式Infinite point：无穷极限Infinite series：无穷级数Integer：整数Integral：积分Integrand：被积分式Integration：积分Intercepts：截距Interval：区间Inverse function：反函数Inverse trigonometric function：反三角函数Iterated integral：逐次积分Intermediate value of Theorem：中间值定理H:Higher mathematics高等数学/高数E、F、G、H:Ellipse：椭圆Ellipsoid：椭圆体Equation：方程式Even function：偶函数Expected Valued：期望值Exponential Function：指数函数Extreme value：极值Focus：焦点Fractions：分式Function：函数Gradient：梯度Graph：图形Higher Derivative：高阶导数Horizontal asymptote：水平渐近线Horizontal line：水平线Hyperbola：双曲线Hyper boloid：双曲面D:Decreasing function：递减函数Decreasing sequence：递减数列Definite integral：定积分Density：密度Derivative：导数higher：高阶导数partial：偏导数Determinant：行列式Differentiable function：可导函数Differential：微分Differential equation：微分方程partial：偏微分方程Differentiation：求导法implicit：隐求导法partial：偏微分法Discontinuity：不连续性Distance：距离Divergence：发散Domain：定义域Double integral：二重积分C:Calculus：微积分differential：微分学integral：积分学Circle：圆Circular cylinder：圆柱Closed interval：封闭区间Coefficient：系数Cone：圆锥Constant function：常数函数Constant of integration：积分常数Continuity：连续性Continuous function：连续函数Convergence：收敛Convergent sequence：收敛数列Coordinate：s：坐标polar：极坐标rectangular：直角坐标spherical：球面坐标Coordinate axes：坐标轴Cosine function：余弦函数Critical point：临界点Cubic function：三次函数Curve：曲线Cylinder：圆柱A、B:Absolute convergence：绝对收敛Absolute extreme values：绝对极值Absolute maximum and minimum：绝对极大与极小Absolute value：绝对值Absolute value function：绝对值函数Acceleration：加速度Antiderivative：反导Arbitrary constant：任意常数Arc length：弧长Area：面积Asymptote：渐近线horizontal：水平渐近线slant：斜渐近线vertical：垂直渐近线Average speed：平均速率Average velocity：平均速度微积分词汇第一章函数与极限Chapter1Function and Limit集合set元素element子集subset空集empty set并集union交集intersection 差集difference of set基本集basic set补集complement set直积direct product开区间open interval闭区间closed interval映射mapping一一映射one-to-one mapping变化transformation函数function自变量independent variable因变量dependent variable函数关系function relation值域range函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part分段函数piecewise function函数的单调性monotonicity of a function单调增加的increasing单调减少的decreasing单调函数monotone function对称symmetry偶函数even function奇函数odd function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inversetrigonometric function常数函数constant function极限limit数列sequence of number 收敛convergence发散divergent子列subsequence函数的极限limits of functions左极限left limit右极限right limit单侧极限one-sided limits无穷小infinitesimal无穷大infinity单调数列monotonic sequence高阶无穷小infinitesimal of higher order 低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order2高等数学-翻译等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind定义区间defined interval最大值global maximum value(absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem 第二章导数与微分Chapter2Derivative and Differential匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line 位置函数position function导数derivative可导derivable导函数derived function切线方程tangent equation隐函数implicit function显函数explicit function高阶导数nth derivative相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3MeanValue Theorem of Differentials and the Application of Derivatives临界点critical point辅助函数auxiliary function不定式indeterminate form泰勒公式Taylor formula余项remainder term拐点inflection point函数的极值extremum of function极大值local(relative)maximum极小值local(relative)minimum曲率curvature平均曲率average curvature曲率中心center of curvature第四章不定积分Chapter4Indefinite Integrals原函数primitive function(antiderivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function第五章定积分Chapter5Definite Integrals曲边梯形trapezoid with曲边curve edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable反常积分improper integral第六章定积分的应用面积元素element of area极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation曲线的弧长arc length of acurve光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7Space Analytic Geometry and Vector Algebra相等equal平行parallel三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law差difference分配律distributive law球面sphere轴axis顶点vertex抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve投影projection垂直perpendicular第八章多元函数微分法及其应用Chapter8Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables边界点frontier point,boundary point开集openset闭集closed set有界集bounded set 无界集unbounded set二重极限double limit连续函数continuous function不连续点discontinuity point偏导数partial derivative高阶偏导数partial derivative of higher order二阶偏导数second order partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition全导数total derivative法线normal line梯度gradient 无条件极值unconditional extreme values条件极值conditional extreme values最小二乘法method of least squares第九章重积分Chapter9Multiple Integrals二重积分double integral可加性additivity三重积分triple integral反常二重积分improper double integral曲面的面积area of a surface质心centre of mass密度density第十二章微分方程Chapter12Differential Equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE初始条件initial condition衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation非齐次线性方程non-homogeneous linear equation全微分方程total differential equation高阶微分方程differential equation of higher order二阶线性微分方程second order linera differential equation线性相关linearly dependence 线性无关linearly independce variable coefficient微分方程组system of differential equations。
计算力学第三章
varying field variable in terms of a finite number of values evaluated at element nodes, for the flexure element, the field variable of interest is the transverse displacement v( x ) of the neutral surface away from its straight, undeflected position. As depicted in Figure 3.2a and 3.2b, the end deflections can be identical, as illustrated, while the deflected shape of the two cases is quite different. Therefore, the flexure element formulation must take into account the slope (rotation) of the beam as well as end-point displacement. In addition to avoiding the potential ambiguity of displacements, inclusion of beam element nodal rotations ensures compatibility of rotations at nodal connections between elements, thus precluding the physically unacceptable discontinuity depicted in Figure 3.2c.
L11中间值定理的应用31derivateegShowthat3x^2x5hasa
L11中間值定理的應用3.1 derivateQ:中間值的內容敘述A:如果有一個函數在AB閉區間上連續，對任意c在f(a)與f(b)之間，則會存在有一個x0在AB閉區間，使得f(x0)=c。
它真正建模的概念f(a) and f(b) 之間的值必被取。
eg. Show that 3x^3-2x+5 has a root.Q:這裡提到AB閉區間了沒？A:沒。
Q:那怎麼辦？A:所以要自己找。
Q:這個題目是要找一個數？A:取值為0。
Q:找怎樣的區間，有沒有條件？A:有。
第一個f在這個閉區間連續。
第二個要使得0被取，0要介於f(a) and f (b)之間。
∵f(0)=5, f(-2)=-24+4-5<0 ∴0 is between f(0) and f(-2).∵f is cont. on [-2,0] ∴By Intermediate value thm. ∃ x0∈[-2,0] s.t f(x0)=0. Therefore f has a root.eg. Let f be cont. on [0,1] and 0≤f(x)≤1. Show that ∃ c∈[0,1] s.t. f(c)=cpf: If f(0)=0 or f(1)=1, then it’s done! 證得，解題的經驗。
assume then f(0)>0 and f(1)<1. 假設f(0)=0 or f(1)=1不發生，也成立。
f(0)=0不發生，所以f(0)>0；f(1)=1不發生，所以f(1)<1Let g(x)=f(x)-x Q:在數學上如何求交集？A:求兩個函數相減有零根∵f and x are cont. on [0,1] ∴g is cont. on [0,1] 根據連續四則運算∵g(0)=f(0)-0>0 and g(1)=f(1)-1<0 ∴0 is between g(0) and g(1).⇒g(c)=f(c)-c=0⇒f(c)=cL11中間值定理的應用 3.1 derivateChapter 3 Differentiation § 3.1 The derivativeLet f be a functionQuestion:How to find the tangent line of graphof the function y=f(x) at (x,f(x))?這個問題就是研究微分的出處 所有人的經驗都說我會畫，但是問題是常常畫壞，變成割線，所以事實上畫割線。
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Economics Suppose c(x) is the total cost of producing x units of a certain commodity in a company. The function c(x) is called a cost function. The marginal cost function is the derivative, c’(x), of the cost function. Economists defined the marginal cost at a production level x to be c(x+1)-c(x), which is the cost of producing one additional unit of the commodity. Since
f ( x h) g ( x h) f ( x ) g ( x ) F ( x h) F ( x ) lim F ( x) lim h0 h0 h h g ( x h) g ( x ) f ( x h) f ( x) f ( x ) g ( x h) lim h0 h h
1 1 1 1 3 1 f (t ) t 2 t 2 (1 3t )t 2 2 2 2
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The Quotient Rule If f and g are both differentiable, then f ( x) f ( x) g ( x) f ( x) g ( x) [ ] , provided g ( x) 0. 2 g ( x) g ( x)
CHAPTER 3 DIFFERENTIATION RULES
3.1 Derivatives of Polynomials and Exponential Functions 3.2 The Product and Quotient Rules 3.3 Rates of Change in the Natural and Social Sciences 3.4 Derivatives of Trigonometric Functions
i 0
4
Example 2 1 2 2 21 (a ) If f ( x) 2 , then f ( x) ( x ) ( 2) x 3 x x 2 2 1 1 2 2 (b) If y 3 x 2 , then y ( x 3 ) x 3 x 3 3 3 1 1 1 1 1 1 1 2 2 2 (c ) ( x ) ( x ) x x 2 2 2 x New Derivatives from Old
15
c( x 1) c( x) c ( x h) c ( x ) c( x 1) c( x) lim c( x), h 0 1 h
it follows that the marginal cost c’(x) at the production level x is approximately the cost of producing the (x+1)st unit.
f ( x) g ( x) f ( x) g ( x)
( fgh) f gh fgh fgh
10
Example 5 Differentiate the function f (t ) t (1 t ).
Solution Using the Product Rule, we have
f ( x) . We have Proof Let F ( x) g ( x) f ( x h) f ( x ) g ( x h) g ( x ) F ( x h) F ( x ) lim F ( x) lim h0 h 0 h h
f ( x h) f ( x) g ( x) f ( x) h lim
3
Example 1
(a) If f ( x) x 5 , then f ( x) 5 x 4 (b) If y x1000 , then y 1000 x 999 d 4 r 3 (c ) ( ) 4 r 2 dr 3
In fact, we will show in Section 3.8 that it is true for all real numbers n. d ( x n ) ( x n ) nx n1 n dx n i n i i (a b) Cn a b where n is any real number.
If f and g are both differentiable, then (1) [cf ( x)] cf ( x) (c constant) (2) [ f ( x) g ( x)] f ( x) g ( x)
5
4 2 y x 6 x 4 Example 3 Find the points on the curve
8
Notice also that, for x > 0, f’(x) is positive and f is increasing. When x < 0, f’(x) is negative and f is decreasing.
f
f’ Figure 2
9
The Product Rule If f and g are both differentiable, then [ f ( x) g ( x)] f ( x) g ( x) f ( x) g ( x) Proof Let F ( x) f ( x) g ( x). We have
Figure 3
14
3.3 Rates of Change in the Natural and Social Sciences Let’s recall from section 2.7 the basic idea behind rates of change. If x changes from x1 to x2, then the change in x is x x2 x1 and the corresponding change in y is y f ( x2 ) f ( x1 ). The difference quotient y f ( x2 ) f ( x1 ) x x2 x1 is the average rate of change of y with respect to x over the interval [x1, x2]. Its limit as x 0 is the derivative f ( x1 ).
2
3.1 Derivatives of Polynomials and Exponential Functions The Power Rule If n is a positive integer, then d n n ( x ) ( x ) nx n1 dx Proof Let f(x) = xn. We can use the Binomial Theorem to do so: f ( x h) f ( x ) ( x h) n x n f ( x) lim lim h 0 h 0 h h n(n 1) n2 2 [ x n nx n1h x h nxh n1 h n ] x n 2 lim h 0 h n(n 1) n2 n 1 lim[nx x h nxh n2 h n1 ] h 0 2 nx n1
h0
g ( x h) g ( x ) f ( x) g ( x) f ( x) g ( x) g 2 ( x)
g ( x h) g ( x) h
12
C C g ( x) 2 g ( x) g ( x)
(c constant)
Example 5 Find an equation of the tangent line to the curve
13
So the slope of the tangent line at (1, e / 2) is y(1) 0. This means that the tangent line at (1, e / 2) horizontal and its equation is y e / 2. [See Figure 3. Notice that the function is increasing and crosses its tangent line at (1, e / 2). ]
y பைடு நூலகம் x /(1 x 2 )
at the point (1, e / 2). Solution Using the Quotient Rule, we have (e x )(1 x 2 ) e x (1 x 2 ) y (1 x 2 ) 2
e x (1 x 2 ) e x (2 x) e x (1 x) 2 2 2 (1 x ) (1 x 2 ) 2
1
3.5 The Chain Rule 3.6 Implicit Differentiation 3.7 Higher Derivatives 3.8 derivatives of Logarithmic Functions 3.9 Hyperbolic Functions 3.10 Related Rates 3.11 Linear Approximations and Differentials
Solution we have f ( x) e x 1. The function f and its derivative f’ are graphed in Figure 2. Notice that f has a horizontal tangent when x = 0;this corresponds to the fact f’(0)=0.